import { type PracticeQuestion } from "../../types/lesson";

export const LINES_ANGLES_EASY: PracticeQuestion[] = [
  {
    id: "087cdcfd",
    type: "mcq",
    questionHtml:
      "In the figure, three lines intersect at point P. If <strong>x = 65</strong> and <strong>y = 75</strong>, what is the value of z ?",
    choices: [
      { label: "A", text: "140" },
      { label: "B", text: "80" },
      { label: "C", text: "40" },
      { label: "D", text: "20" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. The angle that is shown as lying between the y° angle and the z° angle is a vertical angle with the x° angle. Since vertical angles are congruent and <strong>x = 65</strong>, the angle between the y° angle and the z° angle measures 65°. Since the 65° angle, the y° angle, and the z° angle are adjacent and form a straight angle, it follows that the sum of the measures of these three angles is 180°, which is represented by the equation <strong>65 ° + y ° + z ° = 180 °</strong>. It’s given that y = 75. Substituting 75 for y yields <strong>65 ° + 75 ° + z ° = 180 °</strong>, which can be rewritten as <strong>140 ° + z ° = 180 °</strong>. Subtracting 140° from both sides of this equation yields <strong>z ° = 40 °</strong>. Therefore, <strong>z = 40</strong>.Choice A is incorrect and may result from finding the value of <strong>x + y</strong> rather than z. Choices B and D are incorrect and may result from conceptual or computational errors.",
    hasFigure: true,
    figureUrl: "/practice-images/087cdcfd_img1.png",
  },
  {
    id: "0bb39de4",
    type: "mcq",
    questionHtml:
      "Triangles <strong>A B C</strong> and <strong>D E F</strong> are congruent, where <strong>A</strong> corresponds to <strong>D</strong>, and <strong>B</strong> and <strong>E</strong> are right angles. The measure of angle <strong>A</strong> is <strong>18 °</strong>. What is the measure of angle <strong>F</strong>?",
    choices: [
      { label: "A", text: "<strong>18 °</strong>" },
      { label: "B", text: "<strong>72 °</strong>" },
      { label: "C", text: "<strong>90 °</strong>" },
      { label: "D", text: "<strong>162 °</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. It’s given that triangle <strong>A B C</strong> is congruent to triangle <strong>D E F</strong>. Corresponding angles of congruent triangles are congruent and, therefore, have equal measure. It’s given that angle <strong>A</strong> corresponds to angle <strong>D</strong>, and that the measure of angle <strong>A</strong> is <strong>18 °</strong>. It's also given that the measures of angles <strong>B</strong> and <strong>E</strong> are <strong>90 °</strong>. Since these angles have equal measure, they are corresponding angles. It follows that angle <strong>C</strong> corresponds to angle <strong>F</strong>. Let <strong>x °</strong> represent the measure of angle <strong>C</strong>. Since the sum of the measures of the interior angles of a triangle is <strong>180 °</strong>, it follows that <strong>18 ° + 90 ° + x ° = 180 °</strong>, or <strong>108 ° + x ° = 180 °</strong>. Subtracting <strong>108 °</strong> from both sides of this equation yields <strong>x ° = 72 °</strong>. Therefore, the measure of angle <strong>C</strong> is <strong>72 °</strong>. Since angle <strong>C</strong> corresponds to angle <strong>F</strong>, it follows that the measure of angle <strong>F</strong> is also <strong>72 °</strong>.<br>Choice A is incorrect. This is the measure of angle <strong>D</strong>, not the measure of angle <strong>F</strong>.<br>Choice C is incorrect. This is the measure of angle <strong>E</strong>, not the measure of angle <strong>F</strong>.<br>Choice D is incorrect. This is the sum of the measures of angles <strong>E</strong> and <strong>F</strong>, not the measure of angle <strong>F</strong>.",
    hasFigure: false,
  },
  {
    id: "0d3f51dc",
    type: "mcq",
    questionHtml:
      "Clockwise from top left, the lines are labeled t, m, and n.<br>Line t intersects both line m and line n.<br>At the intersection of line t and line m, 1 angle is labeled clockwise from top left as follows:<br><br>Bottom left: x°<br><br>At the intersection of line t and line n, 1 angle is labeled clockwise from top left as follows:<br><br>Bottom right: 33°<br><br>A note indicates the figure is not drawn to scale.<br><br>In the figure, line <strong>m</strong> is parallel to line <strong>n</strong>, and line <strong>t</strong> intersects both lines. What is the value of <strong>x</strong>?",
    choices: [
      { label: "A", text: "<strong>33</strong>" },
      { label: "B", text: "<strong>57</strong>" },
      { label: "C", text: "<strong>123</strong>" },
      { label: "D", text: "<strong>147</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. It’s given that line <strong>m</strong> is parallel to line <strong>n</strong>, and line <strong>t</strong> intersects both lines. It follows that line <strong>t</strong> is a transversal. When two lines are parallel and intersected by a transversal, exterior angles on the same side of the transversal are supplementary. Thus, <strong>x + 33 = 180</strong>. Subtracting <strong>33</strong> from both sides of this equation yields <strong>x = 147</strong>. Therefore, the value of <strong>x</strong> is <strong>147</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.",
    hasFigure: true,
    figureUrl: "/practice-images/0d3f51dc_svg1.svg",
  },
  {
    id: "3563d76d",
    type: "mcq",
    questionHtml:
      "At a certain time and day, the Washington Monument in Washington, DC, casts a shadow that is 300 feet long. At the same time, a nearby cherry tree casts a shadow that is 16 feet long. Given that the Washington Monument is approximately 555 feet tall, which of the following is closest to the height, in feet, of the cherry tree?",
    choices: [
      { label: "A", text: "10" },
      { label: "B", text: "20" },
      { label: "C", text: "30" },
      { label: "D", text: "35" },
    ],
    correctAnswer: "",
    explanation:
      "Choice C is correct. There is a proportional relationship between the height of an object and the length of its shadow. Let c represent the height, in feet, of the cherry tree. The given relationship can be expressed by the proportion <strong>555 over 300 = c over 16</strong>. Multiplying both sides of this equation by 16 yields <strong>c = 29 . 6</strong>. This height is closest to the value given in choice C, 30.Choices A, B, and D are incorrect and may result from calculation errors.",
    hasFigure: false,
  },
  {
    id: "36200a38",
    type: "mcq",
    questionHtml:
      "<strong>The figure presents a triangle with a horizontal base. The left and right sides of the triangle are both extended past the top vertex of the triangle. The bottom left angle is labeled 70 °, and the bottom right angle is labeled 50 °. The angle above the left side of the triangle and below the extension of the right side of the triangle is labeled x °.</strong>In the figure above, two sides of a triangle are extended. What is the value of x ?",
    choices: [
      { label: "A", text: "110" },
      { label: "B", text: "120" },
      { label: "C", text: "130" },
      { label: "D", text: "140" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. The sum of the interior angles of a triangle is 180°. The measures of the two interior angles of the given triangle are shown. Therefore, the measure of the third interior angle is 180° – 70° – 50° = 60°. The angles of measures x° and 60° are supplementary, so their sum is 180°. Therefore, x = 180 – 60 = 120.Choice A is incorrect and may be the result of misinterpreting x° as supplementary to 70°. Choice C is incorrect and may be the result of misinterpreting x° as supplementary to 50°. Choice D is incorrect and may be the result of a calculation error.",
    hasFigure: true,
    figureUrl: "/practice-images/36200a38_img1.png",
  },
  {
    id: "3828f53d",
    type: "mcq",
    questionHtml:
      "In the figure above, lines m and n are parallel. What is the value of b ?",
    choices: [
      { label: "A", text: "40" },
      { label: "B", text: "50" },
      { label: "C", text: "65" },
      { label: "D", text: "80" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. Given that lines m and n are parallel, the angle marked 130° must be supplementary to the leftmost angle marked a° because they are same-side interior angles. Therefore, 130° + a° = 180°, which yields a = 50°. Lines <strong>l</strong> and m intersect at a right angle, so lines j, <strong>l</strong>, and m form a right triangle where the two acute angles are a° and b°. The acute angles of a right triangle are complementary, so a° + b° = 90°, which yields 50° + b° = 90°, and b = 40.Choice B is incorrect. This is the value of a, not b. Choice C is incorrect and may be the result of dividing 130° by 2. Choice D is incorrect and may be the result of multiplying b by 2.",
    hasFigure: true,
    figureUrl: "/practice-images/3828f53d_img1.png",
  },
  {
    id: "3b4b5b1e",
    type: "spr",
    questionHtml:
      "Clockwise from top left, the 3 lines are labeled t, j, and k.<br>Line t intersects both line j and line k.<br>At the intersection of line t and line j, 2 angles are labeled clockwise from top left as follows:<br><br>Bottom right: 133°<br>Bottom left: x°<br><br>A note indicates the figure is not drawn to scale.<br><br>In the figure, line <strong>j</strong> is parallel to line <strong>k</strong>. What is the value of <strong>x</strong>?",
    choices: [],
    correctAnswer: "47",
    explanation:
      "The correct answer is <strong>47</strong>. Based on the figure, the angle with measure <strong>x °</strong> and the angle with measure <strong>133 °</strong> together form a straight line. Therefore, these two angles are supplementary, so the sum of their measures is <strong>180 °</strong>. It follows that <strong>x + 133 = 180</strong>. Subtracting <strong>133</strong> from both sides of this equation yields <strong>x = 47</strong>.",
    hasFigure: true,
    figureUrl: "/practice-images/3b4b5b1e_svg1.svg",
  },
  {
    id: "410bdbe6",
    type: "mcq",
    questionHtml:
      "In the triangle above, <strong>a = 45</strong>. What is the value of b ?",
    choices: [
      { label: "A", text: "52" },
      { label: "B", text: "59" },
      { label: "C", text: "76" },
      { label: "D", text: "104" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. The sum of the measures of the three interior angles of a triangle is 180°. Therefore, <strong>31 + 2 b + a = 180</strong>. Since it’s given that <strong>a = 45</strong>, it follows that <strong>31 + 2 b + 45 = 180</strong>, or <strong>2 b = 104</strong>. Dividing both sides of this equation by 2 yields <strong>b = 52</strong>.Choice B is incorrect and may result from a calculation error. Choice C is incorrect. This is the value of <strong>a + 31</strong>. Choice D is incorrect. This is the value of <strong>2 b</strong>.",
    hasFigure: true,
    figureUrl: "/practice-images/410bdbe6_img1.png",
  },
  {
    id: "42b4493b",
    type: "mcq",
    questionHtml:
      "In a right triangle, the measure of one of the acute angles is <strong>51 °</strong>. What is the measure, in degrees, of the other acute angle?",
    choices: [
      { label: "A", text: "<strong>6</strong>" },
      { label: "B", text: "<strong>39</strong>" },
      { label: "C", text: "<strong>49</strong>" },
      { label: "D", text: "<strong>51</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. The sum of the measures of the interior angles of a triangle is <strong>180</strong> degrees. Since the triangle is a right triangle, it has one angle that measures <strong>90</strong> degrees. Therefore, the sum of the measures, in degrees, of the remaining two angles is <strong>180 − 90</strong>, or <strong>90</strong>. It’s given that the measure of one of the acute angles in the triangle is <strong>51</strong> degrees. Therefore, the measure, in degrees, of the other acute angle is <strong>90 − 51</strong>, or <strong>39</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect. This is the measure, in degrees, of the acute angle whose measure is given.",
    hasFigure: false,
  },
  {
    id: "5207e508",
    type: "mcq",
    questionHtml:
      "Clockwise from top left, the 3 lines are labeled t, m, and n.<br>Line t intersects both line m and line n.<br>At the intersection of line t and line m, 1 angle is labeled clockwise from top left as follows:<br><br>Bottom left: 170°<br><br>At the intersection of line t and line n, 1 angle is labeled clockwise from top left as follows:<br><br>Bottom left: w°<br><br>A note indicates the figure is not drawn to scale.<br><br>In the figure, line <strong>m</strong> is parallel to line <strong>n</strong>. What is the value of <strong>w</strong>?",
    choices: [
      { label: "A", text: "<strong>17</strong>" },
      { label: "B", text: "<strong>30</strong>" },
      { label: "C", text: "<strong>70</strong>" },
      { label: "D", text: "<strong>170</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. It's given that lines <strong>m</strong> and <strong>n</strong> are parallel. Since line <strong>t</strong> intersects both lines <strong>m</strong> and <strong>n</strong>, it's a transversal. The angles in the figure marked as <strong>170 °</strong> and <strong>w °</strong> are on the same side of the transversal, where one is an interior angle with line <strong>m</strong> as a side, and the other is an exterior angle with line <strong>n</strong> as a side. Thus, the marked angles are corresponding angles. When two parallel lines are intersected by a transversal, corresponding angles are congruent and, therefore, have equal measure. It follows that <strong>w ° = 170 °</strong>. Therefore, the value of <strong>w</strong> is <strong>170</strong>. <br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.",
    hasFigure: true,
    figureUrl: "/practice-images/5207e508_svg1.svg",
  },
  {
    id: "5733ce30",
    type: "mcq",
    questionHtml:
      "In the given figure, <strong>side A C</strong> extends to point D. If the measure of <strong>angle B A C</strong> is equal to the measure of <strong>angle B C A</strong>, what is the value of x ?",
    choices: [
      { label: "A", text: "110" },
      { label: "B", text: "70" },
      { label: "C", text: "55" },
      { label: "D", text: "40" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. Since <strong>angle B C D</strong> and <strong>angle B C A</strong> form a linear pair of angles, their measures sum to 180°. It’s given that the measure of <strong>angle B C D</strong> is 110°. Therefore, <strong>110 ° + angle B C A = 180 °</strong>. Subtracting 110° from both sides of this equation gives the measure of <strong>angle B C A</strong> as 70°. It’s also given that the measure of <strong>angle B A C</strong> is equal to the measure of <strong>angle B C A</strong>. Thus, the measure of <strong>angle B A C</strong> is also 70°. The measures of the interior angles of a triangle sum to 180°. Thus, <strong>70 ° + 70 ° + x ° = 180 °</strong>. Combining like terms on the left-hand side of this equation yields <strong>140 ° + x ° = 180 °</strong>. Subtracting 140° from both sides of this equation yields <strong>x ° = 40 °</strong>, or <strong>x = 40</strong>.Choice A is incorrect. This is the value of the measure of <strong>angle B C D</strong>. Choice B is incorrect. This is the value of the measure of each of the other two interior angles, <strong>angle B C A</strong> and <strong>angle B A C</strong>. Choice C is incorrect and may result from an error made when identifying the relationship between the exterior angle of a triangle and the interior angles of the triangle.",
    hasFigure: true,
    figureUrl: "/practice-images/5733ce30_img1.png",
  },
  {
    id: "992f4e93",
    type: "mcq",
    questionHtml:
      "In the figure above, lines <strong>l</strong> and k are parallel. What is the value of a ?",
    choices: [
      { label: "A", text: "26" },
      { label: "B", text: "64" },
      { label: "C", text: "116" },
      { label: "D", text: "154" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. Since lines <strong>l</strong> and k are parallel, corresponding angles formed by the intersection of line j with lines <strong>l</strong> and k are congruent. Therefore, the angle with measure a° must be the supplement of the angle with measure 64°. The sum of two supplementary angles is 180°, so a = 180 – 64 = 116.Choice A is incorrect and likely results from thinking the angle with measure a° is the complement of the angle with measure 64°. Choice B is incorrect and likely results from thinking the angle with measure a° is congruent to the angle with measure 64°. Choice D is incorrect and likely results from a conceptual or computational error.",
    hasFigure: true,
    figureUrl: "/practice-images/992f4e93_img1.png",
  },
  {
    id: "a6dbad6b",
    type: "mcq",
    questionHtml:
      "<strong>The figure presents 2 parallel lines l and m, which are almost horizontal, with l above m. A from the left on m, moves up and to the right, intersects, and ends above l. Another from the right on m, moves up and to the left, intersects, and ends at the same . as the other line segment, above l. The angle at the . where the two line segments meet is labeled x °. The angle above l and to the left of the right is labeled z °. The angle above m and to the right of the left is labeled y °. The figure is not drawn to scale</strong>In the figure above, lines <strong>l</strong> and m are parallel, <strong>y = 20</strong>, and <strong>z = 60</strong>. What is the value of x ?",
    choices: [
      { label: "A", text: "120" },
      { label: "B", text: "100" },
      { label: "C", text: "90" },
      { label: "D", text: "80" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. Let the measure of the third angle in the smaller triangle be <strong>a, °</strong>. Since lines <strong>l</strong> and m are parallel and cut by transversals, it follows that the corresponding angles formed are congruent. So <strong>a ° = y °, which = 20 °</strong>. The sum of the measures of the interior angles of a triangle is <strong>180 °</strong>, which for the interior angles in the smaller triangle yields <strong>a + x + z = 180</strong>. Given that <strong>z = 60</strong> and <strong>a = 20</strong>, it follows that <strong>20 + x + 60 = 180</strong>. Solving for x gives <strong>x = 180 − 60 − 20</strong>, or <strong>x = 100</strong>.Choice A is incorrect and may result from incorrectly assuming that angles <strong>x + z = 180</strong>. Choice C is incorrect and may result from incorrectly assuming that the smaller triangle is a right triangle, with x as the right angle. Choice D is incorrect and may result from a misunderstanding of the exterior angle theorem and incorrectly assuming that <strong>x = y + z</strong>.",
    hasFigure: true,
    figureUrl: "/practice-images/a6dbad6b_img1.png",
  },
  {
    id: "afa3c48b",
    type: "mcq",
    questionHtml:
      "Clockwise from top left, the 3 lines are labeled l, m, and n.<br>Line l intersects both line m and line n.<br>At the intersection of line l and line m, 1 angle is labeled clockwise from top left as follows:<br><br>Top right: x°<br><br>At the intersection of line l and line n, 1 angle is labeled clockwise from top left as follows:<br><br>Top left: 26°<br><br>A note indicates the figure is not drawn to scale.<br><br>In the figure shown, line <strong>m</strong> is parallel to line <strong>n</strong>. What is the value of <strong>x</strong>?",
    choices: [
      { label: "A", text: "<strong>13</strong>" },
      { label: "B", text: "<strong>26</strong>" },
      { label: "C", text: "<strong>52</strong>" },
      { label: "D", text: "<strong>154</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. The sum of consecutive interior angles between two parallel lines and on the same side of the transversal is <strong>180</strong> degrees. Since it's given that line <strong>m</strong> is parallel to line <strong>n</strong>, it follows that <strong>x + 26 = 180</strong>. Subtracting <strong>26</strong> from both sides of this equation yields <strong>154</strong>. Therefore, the value of <strong>x</strong> is <strong>154</strong>.<br>Choice A is incorrect. This is half of the given angle measure.<br>Choice B is incorrect. This is the value of the given angle measure.<br>Choice C is incorrect. This is twice the value of the given angle measure.",
    hasFigure: true,
    figureUrl: "/practice-images/afa3c48b_svg1.svg",
  },
  {
    id: "c24e1bda",
    type: "mcq",
    questionHtml:
      "Clockwise from top left, the 3 lines are labeled t, m, and n.<br>Line t intersects both line m and line n.<br>At the intersection of line t and line m, 1 angle is labeled clockwise from top left as follows:<br><br>Bottom left: 134°<br><br>At the intersection of line t and line n, 1 angle is labeled clockwise from top left as follows:<br><br>Bottom left: w°<br><br>A note indicates the figure is not drawn to scale. <br><br>In the figure, line <strong>m</strong> is parallel to line <strong>n</strong>. What is the value of <strong>w</strong>?",
    choices: [
      { label: "A", text: "<strong>13</strong>" },
      { label: "B", text: "<strong>34</strong>" },
      { label: "C", text: "<strong>66</strong>" },
      { label: "D", text: "<strong>134</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. It's given that lines <strong>m</strong> and <strong>n</strong> are parallel. Since line <strong>t</strong> intersects both lines <strong>m</strong> and <strong>n</strong>, it's a transversal. The angles in the figure marked as <strong>134 °</strong> and <strong>w °</strong> are on the same side of the transversal, where one is an interior angle with line <strong>m</strong> as a side, and the other is an exterior angle with line <strong>n</strong> as a side. Thus, the marked angles are corresponding angles. When two parallel lines are intersected by a transversal, corresponding angles are congruent and, therefore, have equal measure. It follows that <strong>w ° = 134 °</strong>. Therefore, the value of <strong>w</strong> is <strong>134</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.",
    hasFigure: true,
    figureUrl: "/practice-images/c24e1bda_svg1.svg",
  },
  {
    id: "c8d60e48",
    type: "mcq",
    questionHtml:
      "In the given triangle, <strong>the length of side A, B = the length of side A, C</strong> and <strong>angle A, B C</strong> has a measure of <strong>67 °</strong>. What is the value of x ?",
    choices: [
      { label: "A", text: "36" },
      { label: "B", text: "46" },
      { label: "C", text: "58" },
      { label: "D", text: "70" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. Since <strong>the length of A, B = the length of A, C</strong>, the measures of their corresponding angles, <strong>angle A, B C</strong> and <strong>angle A, C B</strong>, are equal. Since <strong>angle A, B C</strong> has a measure of <strong>67 °</strong>, the measure of <strong>angle A, C B</strong> is also <strong>67 °</strong>. Since the sum of the measures of the interior angles in a triangle is <strong>180 °</strong>, it follows that <strong>67 + 67 + x = 180</strong>, or <strong>134 + x = 180</strong>. Subtracting by 134 on both sides of this equation yields <strong>x = 46</strong>.Choices A, C, and D are incorrect and may result from calculation errors.",
    hasFigure: true,
    figureUrl: "/practice-images/c8d60e48_img1.png",
  },
  {
    id: "cbe8ca31",
    type: "mcq",
    questionHtml:
      "In <strong>triangle X Y Z</strong>, the measure of <strong>angle X</strong> is <strong>24 °</strong> and the measure of <strong>angle Y</strong> is <strong>98 °</strong>. What is the measure of <strong>angle Z</strong>?",
    choices: [
      { label: "A", text: "<strong>58 °</strong>" },
      { label: "B", text: "<strong>74 °</strong>" },
      { label: "C", text: "<strong>122 °</strong>" },
      { label: "D", text: "<strong>212 °</strong>" },
    ],
    correctAnswer: "A",
    explanation:
      "Choice A is correct. The triangle angle sum theorem states that the sum of the measures of the interior angles of a triangle is <strong>180 °</strong>. It's given that in <strong>triangle X Y Z</strong>, the measure of <strong>angle X</strong> is <strong>24 °</strong> and the measure of <strong>angle Y</strong> is <strong>98 °</strong>. It follows that the measure of <strong>angle Z</strong> is <strong>(180 − 24 − 98) °</strong>, or <strong>58 °</strong>.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect. This is the sum of the measures of <strong>angle X</strong> and <strong>angle Y</strong>, not the measure of <strong>angle Z</strong>.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "cf0d3050",
    type: "spr",
    questionHtml:
      "Clockwise from top left, the 3 lines are labeled, c, s, and t.<br>Line c intersects both line s and line t.<br>At the intersection of line c and line s, 1 angle is labeled clockwise from top left as follows:<br><br>Top right: x°<br><br>At the intersection of line c and line t, 1 angle is labeled clockwise from top left as follows:<br><br>Top left: 110°<br><br>A note indicates the figure is not drawn to scale.<br><br>In the figure shown, line <strong>c</strong> intersects parallel lines <strong>s</strong> and <strong>t</strong>. What is the value of <strong>x</strong>?",
    choices: [],
    correctAnswer: "70",
    explanation:
      "The correct answer is <strong>70</strong>. Based on the figure, the angle with measure <strong>110 °</strong> and the angle vertical to the angle with measure <strong>x °</strong> are same side interior angles. Since vertical angles are congruent, the angle vertical to the angle with measure <strong>x °</strong> also has measure <strong>x °</strong>. It’s given that lines <strong>s</strong> and <strong>t</strong> are parallel. Therefore, same side interior angles between lines <strong>s</strong> and <strong>t</strong> are supplementary. It follows that <strong>x + 110 = 180</strong>. Subtracting <strong>110</strong> from both sides of this equation yields <strong>x = 70</strong>.",
    hasFigure: true,
    figureUrl: "/practice-images/cf0d3050_svg1.svg",
  },
  {
    id: "d5f349b7",
    type: "mcq",
    questionHtml:
      "Right triangle upper P upper Q upper R is labeled as follows:<br><br>Angle upper R is a right angle.<br><br>Right triangle upper S upper T upper U is labeled as follows:<br><br>Angle upper U is a right angle.<br><br>A note indicates the figures are not drawn to scale.<br><br>Right triangles <strong>P Q R</strong> and <strong>S T U</strong> are similar, where <strong>P</strong> corresponds to <strong>S</strong>. If the measure of angle <strong>Q</strong> is <strong>18 °</strong>, what is the measure of angle <strong>S</strong>?",
    choices: [
      { label: "A", text: "<strong>18 °</strong>" },
      { label: "B", text: "<strong>72 °</strong>" },
      { label: "C", text: "<strong>82 °</strong>" },
      { label: "D", text: "<strong>162 °</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. In similar triangles, corresponding angles are congruent. It’s given that right triangles <strong>P Q R</strong> and <strong>S T U</strong> are similar, where angle <strong>P</strong> corresponds to angle <strong>S</strong>. It follows that angle <strong>P</strong> is congruent to angle <strong>S</strong>. In the triangles shown, angle <strong>R</strong> and angle <strong>U</strong> are both marked as right angles, so angle <strong>R</strong> and angle <strong>U</strong> are corresponding angles. It follows that angle <strong>Q</strong> and angle <strong>T</strong> are corresponding angles, and thus, angle <strong>Q</strong> is congruent to angle <strong>T</strong>. It’s given that the measure of angle <strong>Q</strong> is <strong>18 °</strong>, so the measure of angle <strong>T</strong> is also <strong>18 °</strong>. Angle <strong>U</strong> is a right angle, so the measure of angle <strong>U</strong> is <strong>90 °</strong>. The sum of the measures of the interior angles of a triangle is <strong>180 °</strong>. Thus, the sum of the measures of the interior angles of triangle <strong>S T U</strong> is <strong>180</strong> degrees. Let <strong>s</strong> represent the measure, in degrees, of angle <strong>S</strong>. It follows that <strong>s + 18 + 90 = 180</strong>, or <strong>s + 108 = 180</strong>. Subtracting <strong>108</strong> from both sides of this equation yields <strong>s = 72</strong>. Therefore, if the measure of angle <strong>Q</strong> is <strong>18</strong> degrees, then the measure of angle <strong>S</strong> is <strong>72</strong> degrees.<br>Choice A is incorrect. This is the measure of angle <strong>T</strong>.<br>Choice C is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect. This is the sum of the measures of angle <strong>S</strong> and angle <strong>U</strong>.",
    hasFigure: true,
    figureUrl: "/practice-images/d5f349b7_svg1.svg",
  },
  {
    id: "dfc420b2",
    type: "mcq",
    questionHtml:
      "In the figure above, <strong>A, D</strong> intersects <strong>B E</strong> at C. If <strong>x = 100</strong>, what is the value of y ?",
    choices: [
      { label: "A", text: "100" },
      { label: "B", text: "90" },
      { label: "C", text: "80" },
      { label: "D", text: "60" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. It’s given that <strong>x = 100</strong>; therefore, substituting 100 for x in triangle ABC gives two known angle measures for this triangle. The sum of the measures of the interior angles of any triangle equals 180°. Subtracting the two known angle measures of triangle ABC from 180° gives the third angle measure: <strong>180 ° − 100 ° − 20 ° = 60 °</strong>. This is the measure of angle BCA. Since vertical angles are congruent, the measure of angle DCE is also 60°. Subtracting the two known angle measures of triangle CDE from 180° gives the third angle measure: <strong>180 ° − 60 ° − 40 ° = 80 °</strong>. Therefore, the value of y is 80.Choice A is incorrect and may result from a calculation error. Choice B is incorrect and may result from classifying angle CDE as a right angle. Choice D is incorrect and may result from finding the measure of angle BCA or DCE instead of the measure of angle CDE.",
    hasFigure: true,
    figureUrl: "/practice-images/dfc420b2_img1.png",
  },
  {
    id: "f1747a6a",
    type: "mcq",
    questionHtml:
      "In triangle <strong>A B C</strong>, the measure of angle <strong>B</strong> is <strong>52 °</strong> and the measure of angle <strong>C</strong> is <strong>17 °</strong>. What is the measure of angle <strong>A</strong>?",
    choices: [
      { label: "A", text: "<strong>21 °</strong>" },
      { label: "B", text: "<strong>35 °</strong>" },
      { label: "C", text: "<strong>69 °</strong>" },
      { label: "D", text: "<strong>111 °</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. The sum of the angle measures of a triangle is <strong>180 °</strong>. Adding the measures of angles <strong>B</strong> and <strong>C</strong> gives <strong>52 + 17 = 69 °</strong>. Therefore, the measure of angle <strong>A</strong> is <strong>180 − 69 = 111 °</strong>.<br>Choice A is incorrect and may result from subtracting the sum of the measures of angles <strong>B</strong> and <strong>C</strong> from <strong>90 °</strong>, instead of from <strong>180 °</strong>.<br>Choice B is incorrect and may result from subtracting the measure of angle <strong>C</strong> from the measure of angle <strong>B</strong>.<br>Choice C is incorrect and may result from adding the measures of angles <strong>B</strong> and <strong>C</strong> but not subtracting the result from <strong>180 °</strong>.",
    hasFigure: false,
  },
  {
    id: "f9d40000",
    type: "mcq",
    questionHtml:
      "In <strong>triangle X Y Z</strong>, the measure of <strong>angle X</strong> is <strong>23 °</strong> and the measure of <strong>angle Y</strong> is <strong>66 °</strong>. What is the measure of <strong>angle Z</strong>?",
    choices: [
      { label: "A", text: "<strong>43 °</strong>" },
      { label: "B", text: "<strong>89 °</strong>" },
      { label: "C", text: "<strong>91 °</strong>" },
      { label: "D", text: "<strong>179 °</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. The triangle angle sum theorem states that the sum of the measures of the interior angles of a triangle is <strong>180 °</strong>. It's given that in <strong>triangle X Y Z</strong>, the measure of <strong>angle X</strong> is <strong>23 °</strong> and the measure of <strong>angle Y</strong> is <strong>66 °</strong>. It follows that the measure of <strong>angle Z</strong> is <strong>(180 − 23 − 66) °</strong>, or <strong>91 °</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect. This is the sum of the measures of <strong>angle X</strong> and <strong>angle Y</strong>, not the measure of <strong>angle Z</strong>.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
];

export const LINES_ANGLES_MEDIUM: PracticeQuestion[] = [
  {
    id: "1c3d613c",
    type: "spr",
    questionHtml:
      "Triangle ABC and triangle DEF are shown. The relationship between the side lengths of the two triangles is such that <strong>the length of side A, B, over, the length of side D E = the length of side B C, over, the length of side E F, which = the length of side A, C, over, the length of side D F, which = 3</strong>. If the measure of angle BAC is 20°, what is the measure, in degrees, of angle EDF ? (Disregard the degree symbol when gridding your answer.)",
    choices: [],
    correctAnswer: "",
    explanation:
      "The correct answer is 20. By the equality given, the three pairs of corresponding sides of the two triangles are in the same proportion. By the side-side-side (SSS) similarity theorem, triangle ABC is similar to triangle DEF. In similar triangles, the measures of corresponding angles are congruent. Since angle BAC corresponds to angle EDF, these two angles are congruent and their measures are equal. It’s given that the measure of angle BAC is 20°, so the measure of angle EDF is also 20°.",
    hasFigure: true,
    figureUrl: "/practice-images/1c3d613c_img1.png",
  },
  {
    id: "2adbf1b1",
    type: "mcq",
    questionHtml:
      "Clockwise from top left, the 3 lines are labeled t, m, and n.<br>Line t intersects both line m and line n.<br>At the intersection of line t and line m, 2 angles are labeled clockwise from top left as follows:<br><br>Top right: x°<br>Bottom left: y°<br><br>At the intersection of line t and line n, 1 angle is labeled clockwise from top left as follows:<br><br>Top left: z°<br><br>A note indicates the figure is not drawn to scale.<br><br>In the figure, lines <strong>m</strong> and <strong>n</strong> are parallel. If <strong>x = 6 k + 13</strong> and <strong>y = 8 k − 29</strong>, what is the value of <strong>z</strong>?",
    choices: [
      { label: "A", text: "<strong>3</strong>" },
      { label: "B", text: "<strong>21</strong>" },
      { label: "C", text: "<strong>41</strong>" },
      { label: "D", text: "<strong>139</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. Vertical angles, which are angles that are opposite each other when two lines intersect, are congruent. The figure shows that lines <strong>t</strong> and <strong>m</strong> intersect. It follows that the angle with measure <strong>x °</strong> and the angle with measure <strong>y °</strong> are vertical angles, so <strong>x = y</strong>. It's given that <strong>x = 6 k + 13</strong> and <strong>y = 8 k − 29</strong>. Substituting <strong>6 k + 13</strong> for <strong>x</strong> and <strong>8 k − 29</strong> for <strong>y</strong> in the equation <strong>x = y</strong> yields <strong>6 k + 13 = 8 k − 29</strong>. Subtracting <strong>6 k</strong> from both sides of this equation yields <strong>13 = 2 k − 29</strong>. Adding <strong>29</strong> to both sides of this equation yields <strong>42 = 2 k</strong>, or <strong>2 k = 42</strong>. Dividing both sides of this equation by <strong>2</strong> yields <strong>k = 21</strong>. It's given that lines <strong>m</strong> and <strong>n</strong> are parallel, and the figure shows that lines <strong>m</strong> and <strong>n</strong> are intersected by a transversal, line <strong>t</strong>. If two parallel lines are intersected by a transversal, then the same-side interior angles are supplementary. It follows that the same-side interior angles with measures <strong>y °</strong> and <strong>z °</strong> are supplementary, so <strong>y + z = 180</strong>. Substituting <strong>8 k − 29</strong> for <strong>y</strong> in this equation yields <strong>8 k − 29 + z = 180</strong>. Substituting <strong>21</strong> for <strong>k</strong> in this equation yields <strong>8 (21) − 29 + z = 180</strong>, or <strong>139 + z = 180</strong>. Subtracting <strong>139</strong> from both sides of this equation yields <strong>z = 41</strong>. Therefore, the value of <strong>z</strong> is <strong>41</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect. This is the value of <strong>k</strong>, not <strong>z</strong>.<br>Choice D is incorrect. This is the value of <strong>x</strong> or <strong>y</strong>, not <strong>z</strong>.",
    hasFigure: true,
    figureUrl: "/practice-images/2adbf1b1_svg1.svg",
  },
  {
    id: "4ff7b652",
    type: "mcq",
    questionHtml:
      "Right triangles <strong>L M N</strong> and <strong>P Q R</strong> are similar, where <strong>L</strong> and <strong>M</strong> correspond to <strong>P</strong> and <strong>Q</strong>, respectively. Angle <strong>M</strong> has a measure of <strong>53 °</strong>. What is the measure of angle <strong>Q</strong>?",
    choices: [
      { label: "A", text: "<strong>37 °</strong>" },
      { label: "B", text: "<strong>53 °</strong>" },
      { label: "C", text: "<strong>127 °</strong>" },
      { label: "D", text: "<strong>143 °</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. It’s given that triangle <strong>L M N</strong> is similar to triangle <strong>P Q R</strong>. Corresponding angles of similar triangles are congruent. Since angle <strong>M</strong> and angle <strong>Q</strong> correspond to each other, they must be congruent. Therefore, if the measure of angle <strong>M</strong> is <strong>53 °</strong>, then the measure of angle <strong>Q</strong> is also <strong>53 °</strong>.<br>Choice A is incorrect and may result from concluding that angle <strong>M</strong> and angle <strong>Q</strong> are complementary rather than congruent.<br>Choice C is incorrect and may result from concluding that angle <strong>M</strong> and angle <strong>Q</strong> are supplementary rather than congruent.<br>Choice D is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "686b5212",
    type: "spr",
    questionHtml:
      "Clockwise from top left, the 3 lines are labeled s, q, and r.<br>Line s intersects both line q and line r.<br>At the intersection of line s and line q, 1 angle is labeled clockwise from top left as follows:<br><br>Top right: 58°<br><br>At the intersection of line s and line r, 1 angle is labeled clockwise from top left as follows:<br><br>Top left: y°<br><br>A note indicates the figure is not drawn to scale.<br><br>In the figure, line <strong>q</strong> is parallel to line <strong>r</strong>, and both lines are intersected by line <strong>s</strong>. If <strong>y = 2 x + 8</strong>, what is the value of <strong>x</strong>?",
    choices: [],
    correctAnswer: "57",
    explanation:
      "The correct answer is <strong>57</strong>. Based on the figure, the angle with measure <strong>y °</strong> and the angle vertical to the angle with measure <strong>58 °</strong> are same side interior angles. Since vertical angles are congruent, the angle vertical to the angle with measure <strong>58 °</strong> also has measure <strong>58 °</strong>. It’s given that lines <strong>q</strong> and <strong>r</strong> are parallel. Therefore, same side interior angles between lines <strong>q</strong> and <strong>r</strong> are supplementary. It follows that <strong>y + 58 = 180</strong>. If <strong>y = 2 x + 8</strong>, then the value of <strong>x</strong> can be found by substituting <strong>2 x + 8</strong> for <strong>y</strong> in the equation <strong>y + 58 = 180</strong>, which yields <strong>(2 x + 8) + 58 = 180</strong>, or <strong>2 x + 66 = 180</strong>. Subtracting <strong>66</strong> from both sides of this equation yields <strong>2 x = 114</strong>. Dividing both sides of this equation by <strong>2</strong> yields <strong>x = 57</strong>. Thus, if <strong>y = 2 x + 8</strong>, the value of <strong>x</strong> is <strong>57</strong>.",
    hasFigure: true,
    figureUrl: "/practice-images/686b5212_svg1.svg",
  },
  {
    id: "6dd463ca",
    type: "mcq",
    questionHtml:
      "In the figure above, segments AE and BD are parallel. If angle BDC measures 58° and angle ACE measures 62°, what is the measure of angle CAE ?",
    choices: [
      { label: "A", text: "58°" },
      { label: "B", text: "60°" },
      { label: "C", text: "62°" },
      { label: "D", text: "120°" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. It’s given that angle ACE measures <strong>62 °</strong>. Since segments AE and BD are parallel, angles BDC and CEA are congruent. Therefore, angle CEA measures <strong>58 °</strong>. The sum of the measures of angles ACE, CEA, and CAE is <strong>180 °</strong> since the sum of the interior angles of triangle ACE is equal to <strong>180 °</strong>. Let the measure of angle CAE be <strong>x °</strong>. Therefore, <strong>62 + 58 + x = 180</strong>, which simplifies to <strong>x = 60</strong>. Thus, the measure of angle CAE is <strong>60 °</strong>.Choice A is incorrect. This is the measure of angle AEC, not that of angle CAE. Choice C is incorrect. This is the measure of angle ACE, not that of CAE. Choice D is incorrect. This is the sum of the measures of angles ACE and CEA.",
    hasFigure: true,
    figureUrl: "/practice-images/6dd463ca_img1.png",
  },
  {
    id: "81b664bc",
    type: "mcq",
    questionHtml:
      "<strong>The figure presents quadrilateral A, C D F, where side A, F and side C D are horizontal, and side C D lies above side A, F. Side C D is longer than side A, F. . B lies on left side A, C, and . E lies on right side D F. B E, which is horizontal, divides quadrilateral A, C D F into two quadrilaterals, A, B E F and B C D E.</strong>In the figure above, <strong>side A, F</strong>, <strong>B E</strong>, and <strong>side C D</strong> are parallel. Points B and E lie on <strong>side A, C</strong> and <strong>side F D</strong>, respectively. If <strong>the length of A, B = 9, the length of B C = 18 . 5</strong>, and <strong>the length of F E = 8 . 5</strong>, what is the length of <strong>E D</strong>, to the nearest tenth?",
    choices: [
      { label: "A", text: "16.8" },
      { label: "B", text: "17.5" },
      { label: "C", text: "18.4" },
      { label: "D", text: "19.6" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. Since <strong>A F</strong>, <strong>B E</strong>, and <strong>C D</strong> are parallel, quadrilaterals <strong>A F E B</strong> and <strong>B E D C</strong> are similar. Let x represent the length of <strong>E D</strong>. With similar figures, the ratios of the lengths of corresponding sides are equal. It follows that <strong>9 over 18 . 5 = 8 . 5 over x</strong>. Multiplying both sides of this equation by 18.5 and by x yields <strong>9 x = 18 . 5 · 8 . 5</strong>, or <strong>9 x = 157 . 2 5</strong>. Dividing both sides of this equation by 9 yields <strong>x = 17 . 4 7</strong>, which to the nearest tenth is 17.5.Choices A, C, and D are incorrect and may result from errors made when setting up the proportion.",
    hasFigure: true,
    figureUrl: "/practice-images/81b664bc_img1.png",
  },
  {
    id: "901e3285",
    type: "mcq",
    questionHtml:
      "In triangle ABC, the measure of angle A is <strong>50 °</strong>. If triangle ABC is isosceles, which of the following is NOT a possible measure of angle B ?",
    choices: [
      { label: "A", text: "<strong>50 °</strong>" },
      { label: "B", text: "<strong>65 °</strong>" },
      { label: "C", text: "<strong>80 °</strong>" },
      { label: "D", text: "<strong>100 °</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. The sum of the three interior angles in a triangle is <strong>180 °</strong>. It’s given that angle A measures <strong>50 °</strong>. If angle B measured <strong>100 °</strong>, the measure of angle C would be <strong>180 ° − (50 ° + 100 °, ) = 30 °</strong>. Thus, the measures of the angles in the triangle would be <strong>50 °</strong>, <strong>100 °</strong>, and <strong>30 °</strong>. However, an isosceles triangle has two angles of equal measure. Therefore, angle B can’t measure <strong>100 °</strong>.Choice A is incorrect. If angle B has measure <strong>50 °</strong>, then angle C would measure <strong>180 ° − (50 ° + 50 °, ) = 80 °</strong>, and <strong>50 °</strong>, <strong>50 °</strong>, and <strong>80 °</strong> could be the angle measures of an isosceles triangle. Choice B is incorrect. If angle B has measure <strong>65 °</strong>, then angle C would measure <strong>180 ° − (65 ° + 50 °, ) = 65 °</strong>, and <strong>50 °</strong>, <strong>65 °</strong>, and <strong>65 °</strong> could be the angle measures of an isosceles triangle. Choice C is incorrect. If angle B has measure <strong>80 °</strong>, then angle C would measure <strong>180 ° − (80 ° + 50 °, ) = 50 °</strong>, and <strong>50 °</strong>, <strong>80 °</strong>, and <strong>50 °</strong> could be the angle measures of an isosceles triangle.",
    hasFigure: false,
  },
  {
    id: "933fee1a",
    type: "mcq",
    questionHtml:
      "Triangles ABC and DEF are shown above. Which of the following is equal to the ratio <strong>B C over A, B</strong>?",
    choices: [
      { label: "A", text: "<strong>D E over D F</strong>" },
      { label: "B", text: "<strong>D F over D E</strong>" },
      { label: "C", text: "<strong>D F over E F</strong>" },
      { label: "D", text: "<strong>E F over D E</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. In right triangle ABC, the measure of angle B must be 58° because the sum of the measure of angle A, which is 32°, and the measure of angle B is 90°. Angle D in the right triangle DEF has measure 58°. Hence, triangles ABC and DEF are similar (by angle-angle similarity). Since <strong>side B C</strong> is the side opposite to the angle with measure 32° and AB is the hypotenuse in right triangle ABC, the ratio <strong>the length of side B C over the length of side A, B</strong> is equal to <strong>the length of side D F over the length of side D E</strong>.Alternate approach: The trigonometric ratios can be used to answer this question. In right triangle ABC, the ratio <strong>the length of side B C over the length of side A, B = the sin of 32 °</strong>. The angle E in triangle DEF has measure 32° because <strong>the measure of angle D + the measure of angle E = 90 °</strong>. In triangle DEF, the ratio <strong>the length of side D F over the length of side D E = the sin of 32 °</strong>. Therefore, <strong>the length of side D F over the length of side D E = the length of side B C over the length of side A, B</strong>.<br>Choice A is incorrect because <strong>the length of side D E over the length of side D F</strong> is the reciprocal of the ratio <strong>the length of side B C over the length of side A, B</strong>. Choice C is incorrect because <strong>the length of side D F over the length of side D E = the length of side B C over the length of side A, C</strong>, not <strong>the length of side B C over the length of side A, B</strong>. Choice D is incorrect because <strong>the length of side E F over the length of side D E = the length of side A, C over the length of side A, B</strong>, not <strong>the length of side B C over the length of side A, B</strong>.",
    hasFigure: true,
    figureUrl: "/practice-images/933fee1a_img1.png",
  },
  {
    id: "94364a79",
    type: "mcq",
    questionHtml:
      "Two nearby trees are perpendicular to the ground, which is flat. One of these trees is <strong>10</strong> feet tall and has a shadow that is <strong>5</strong> feet long. At the same time, the shadow of the other tree is <strong>2</strong> feet long. How tall, in feet, is the other tree?",
    choices: [
      { label: "A", text: "<strong>3</strong>" },
      { label: "B", text: "<strong>4</strong>" },
      { label: "C", text: "<strong>8</strong>" },
      { label: "D", text: "<strong>27</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. Each tree and its shadow can be modeled using a right triangle, where the height of the tree and the length of its shadow are the legs of the triangle. At a given point in time, the right triangles formed by two nearby trees and their respective shadows will be similar. Therefore, if the height of the other tree is <strong>x</strong>, in feet, the value of <strong>x</strong> can be calculated by solving the proportional relationship <strong>(10 feet tall) / (5 feet long) = (x feet tall) / (2 feet long)</strong>. This equation is equivalent to<strong>(10) / (5) = (x) / (2)</strong>, or <strong>2 = (x) / (2)</strong>. Multiplying each side of the equation <strong>2 = (x) / (2)</strong> by <strong>2</strong> yields <strong>4 = x</strong>. Therefore, the other tree is <strong>4 feet</strong> tall.<br>Choice A is incorrect and may result from calculating the difference between the lengths of the shadows, rather than the height of the other tree.<br>Choice C is incorrect and may result from calculating the difference between the height of the <strong>10</strong>-foot-tall tree and the length of the shadow of the other tree, rather than calculating the height of the other tree.<br>Choice D is incorrect and may result from a conceptual or calculation error.",
    hasFigure: false,
  },
  {
    id: "a4c05a1b",
    type: "mcq",
    questionHtml:
      "Clockwise from top left, the 3 lines are labeled m, r, and s.<br>Line m intersects both line r and line s.<br>At the intersection of line m and line r, 1 angle is labeled clockwise from top left as follows:<br><br>Top right: x°<br><br>At the intersection of line m and line s, 1 angle is labeled clockwise from top left as follows:<br><br>Bottom right: y°<br><br>A note indicates the figure is not drawn to scale.<br><br>In the figure shown, lines <strong>r</strong> and <strong>s</strong> are parallel, and line <strong>m</strong> intersects both lines. If <strong>y < 65</strong>, which of the following must be true?",
    choices: [
      { label: "A", text: "<strong>x < 115</strong>" },
      { label: "B", text: "<strong>x > 115</strong>" },
      { label: "C", text: "<strong>x + y < 180</strong>" },
      { label: "D", text: "<strong>x + y > 180</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. In the figure shown, the angle measuring <strong>y °</strong> is congruent to its vertical angle formed by lines <strong>s</strong> and <strong>m</strong>, so the measure of the vertical angle is also <strong>y °</strong>. The vertical angle forms a same-side interior angle pair with the angle measuring <strong>x °</strong>. It's given that lines <strong>r</strong> and <strong>s</strong> are parallel. Therefore, same-side interior angles in the figure are supplementary, which means the sum of the measure of the vertical angle and the measure of the angle measuring <strong>x °</strong> is <strong>180 °</strong>, or <strong>x + y = 180</strong>. Subtracting <strong>x</strong> from both sides of this equation yields <strong>y = 180 − x</strong>. Substituting <strong>180 − x</strong> for <strong>y</strong> in the inequality <strong>y < 65</strong> yields <strong>180 − x < 65</strong>. Adding <strong>x</strong> to both sides of this inequality yields <strong>180 < 65 + x</strong>. Subtracting <strong>65</strong> from both sides of this inequality yields <strong>115 < x</strong>, or <strong>x > 115</strong>. Thus, if <strong>y < 65</strong>, it must be true that <strong>x > 115</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect. <strong>x + y</strong> must be equal to, not less than, <strong>180</strong>.<br>Choice D is incorrect. <strong>x + y</strong> must be equal to, not greater than, <strong>180</strong>.",
    hasFigure: true,
    figureUrl: "/practice-images/a4c05a1b_svg1.svg",
  },
  {
    id: "c7bed21d",
    type: "mcq",
    questionHtml:
      "Quadrilateral <strong>P prime Q prime R prime S prime</strong> is similar to quadrilateral <strong>P Q R S</strong>, where <strong>P</strong>, <strong>Q</strong>, <strong>R</strong>, and <strong>S</strong> correspond to <strong>P prime</strong>, <strong>Q prime</strong>, <strong>R prime</strong>, and <strong>S prime</strong>, respectively. The measure of angle <strong>P</strong> is <strong>30 °</strong>, the measure of angle <strong>Q</strong> is <strong>50 °</strong>, and the measure of angle <strong>R</strong> is <strong>70 °</strong>. The length of each side of <strong>P prime Q prime R prime S prime</strong> is <strong>3</strong> times the length of each corresponding side of <strong>P Q R S</strong>. What is the measure of angle <strong>P prime</strong>?",
    choices: [
      { label: "A", text: "<strong>10 °</strong>" },
      { label: "B", text: "<strong>30 °</strong>" },
      { label: "C", text: "<strong>40 °</strong>" },
      { label: "D", text: "<strong>90 °</strong>" },
    ],
    correctAnswer: "B",
    explanation:
      "Choice B is correct. It's given that quadrilateral <strong>P prime Q prime R prime S prime</strong> is similar to quadrilateral <strong>P Q R S</strong>, where <strong>P</strong>, <strong>Q</strong>, <strong>R</strong>, and <strong>S</strong> correspond to <strong>P prime</strong>, <strong>Q prime</strong>, <strong>R prime</strong>, and <strong>S prime</strong>, respectively. Since corresponding angles of similar quadrilaterals are congruent, it follows that the measure of angle <strong>P</strong> is equal to the measure of angle <strong>P prime</strong>. It's given that the measure of angle <strong>P</strong> is <strong>30 °</strong>. Therefore, the measure of angle <strong>P prime</strong> is <strong>30 °</strong>.<br>Choice A is incorrect. This is <strong>one third</strong> the measure of angle <strong>P prime</strong>.<br>Choice C is incorrect and may result from conceptual or calculation errors.<br>Choice D is incorrect. This is <strong>3</strong> times the measure of angle <strong>P prime</strong>.",
    hasFigure: false,
  },
  {
    id: "d3fe472f",
    type: "mcq",
    questionHtml:
      "Triangle <strong>A B C</strong> is similar to triangle <strong>X Y Z</strong>, such that <strong>A</strong>, <strong>B</strong>, and <strong>C</strong> correspond to <strong>X</strong>, <strong>Y</strong>, and <strong>Z</strong> respectively. The length of each side of triangle <strong>X Y Z</strong> is <strong>2</strong> times the length of its corresponding side in triangle <strong>A B C</strong>. The measure of side <strong>A B</strong> is <strong>16</strong>. What is the measure of side <strong>X Y</strong>?",
    choices: [
      { label: "A", text: "<strong>14</strong>" },
      { label: "B", text: "<strong>16</strong>" },
      { label: "C", text: "<strong>18</strong>" },
      { label: "D", text: "<strong>32</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. It's given that triangle <strong>A B C</strong> is similar to triangle <strong>X Y Z</strong>, such that <strong>A</strong>, <strong>B</strong>, and <strong>C</strong> correspond to <strong>X</strong>, <strong>Y</strong>, and <strong>Z</strong>, respectively. Therefore, side <strong>A B</strong> corresponds to side <strong>X Y</strong>. Since the length of each side of triangle <strong>X Y Z</strong> is <strong>2</strong> times the length of its corresponding side in triangle <strong>A B C</strong>, it follows that the measure of side <strong>X Y</strong> is <strong>2</strong> times the measure of side <strong>A B</strong>. Thus, since the measure of side <strong>A B</strong> is <strong>16</strong>, then the measure of side <strong>X Y</strong> is <strong>2 (16)</strong>, or <strong>32</strong>.<br>Choice A is incorrect and may result from conceptual or calculation errors.<br>Choice B is incorrect. This is the measure of side <strong>A B</strong>, not side <strong>X Y</strong>.<br>Choice C is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "fd8745fc",
    type: "spr",
    questionHtml:
      "In triangle <strong>J K L</strong>, the measures of <strong>angle K</strong> and <strong>angle L</strong> are each <strong>48 °</strong>. What is the measure of <strong>angle J</strong>, in degrees? (Disregard the degree symbol when entering your answer.)",
    choices: [],
    correctAnswer: "84",
    explanation:
      "The correct answer is <strong>84</strong>. The sum of the measures of the interior angles of a triangle is <strong>180 °</strong>. It's given that in triangle <strong>J K L</strong>, the measures of <strong>angle K</strong>and <strong>angle L</strong> are each <strong>48 °</strong>. Adding the measures, in degrees, of <strong>angle K</strong> and <strong>angle L</strong> gives <strong>48 + 48</strong>, or <strong>96</strong>. Therefore, the measure of <strong>angle J</strong>, in degrees, is <strong>180 − 96</strong>, or <strong>84</strong>.",
    hasFigure: false,
  },
];

export const LINES_ANGLES_HARD: PracticeQuestion[] = [
  {
    id: "17912810",
    type: "spr",
    questionHtml:
      "Counterclockwise from top right, the lines are labeled s, r, q, and t.<br>Line s intersects line r above line q.<br>Line s and line r each intersect line q and line t.<br>At the intersection of line s and line r, 1 angle is labeled counterclockwise from top as follows:<br><br>Top: a°<br><br>At the intersection of line s and line q, 1 angle is labeled counterclockwise from top-left as follows:<br><br>Top left: b°<br><br>At the intersection of line r and line t:<br><br>The line segment divides the bottom-left angle into two equal angles, each labeled w°.<br><br>A note indicates the figure is not drawn to scale.<br><br>In the figure, parallel lines <strong>q</strong> and <strong>t</strong> are intersected by lines <strong>r</strong> and <strong>s</strong>. If <strong>a = 43</strong> and <strong>b = 122</strong>, what is the value of <strong>w</strong>?",
    choices: [],
    correctAnswer: "101/2, 50.5",
    explanation:
      "The correct answer is <strong>(101) / (2)</strong>. In the figure, lines <strong>q</strong>, <strong>r</strong>, and <strong>s</strong> form a triangle. One interior angle of this triangle is vertical to the angle marked <strong>a °</strong>; therefore, the interior angle also has measure <strong>a °</strong>. It's given that <strong>a = 43</strong>. Therefore, the interior angle of the triangle has measure <strong>43 °</strong>. A second interior angle of the triangle forms a straight line, <strong>q</strong>, with the angle marked <strong>b °</strong>. Therefore, the sum of the measures of these two angles is <strong>180 °</strong>. It's given that <strong>b = 122</strong>. Therefore, the angle marked <strong>b °</strong> has measure <strong>122 °</strong> and the second interior angle of the triangle has measure <strong>(180 − 122) °</strong>, or <strong>58 °</strong>. The sum of the interior angles of a triangle is <strong>180 °</strong>. Therefore, the measure of the third interior angle of the triangle is <strong>(180 − 43 − 58) °</strong>, or <strong>79 °</strong>. It's given that parallel lines <strong>q</strong> and <strong>t</strong> are intersected by line <strong>r</strong>. It follows that the triangle's interior angle with measure <strong>79 °</strong> is congruent to the same side interior angle between lines <strong>q</strong> and <strong>t</strong> formed by lines <strong>t</strong> and <strong>r</strong>. Since this angle is supplementary to the two angles marked <strong>w °</strong>, the sum of <strong>79 °</strong>, <strong>w °</strong>, and <strong>w °</strong> is <strong>180 °</strong>. It follows that <strong>79 + w + w = 180</strong>, or <strong>79 + 2 w = 180</strong>. Subtracting <strong>79</strong> from both sides of this equation yields <strong>2 w = 101</strong>. Dividing both sides of this equation by <strong>2</strong> yields <strong>w = (101) / (2)</strong>. Note that 101/2 and 50.5 are examples of ways to enter a correct answer.",
    hasFigure: true,
    figureUrl: "/practice-images/17912810_svg1.svg",
  },
  {
    id: "3b225698",
    type: "mcq",
    questionHtml:
      "Triangle <strong>X Y Z</strong> is similar to triangle <strong>R S T</strong> such that <strong>X</strong>, <strong>Y</strong>, and <strong>Z</strong> correspond to <strong>R</strong>, <strong>S</strong>, and <strong>T</strong>, respectively. The measure of <strong>angle Z</strong> is <strong>20 °</strong> and <strong>2 X Y = R S</strong>. What is the measure of <strong>angle T</strong>?",
    choices: [
      { label: "A", text: "<strong>2 °</strong>" },
      { label: "B", text: "<strong>10 °</strong>" },
      { label: "C", text: "<strong>20 °</strong>" },
      { label: "D", text: "<strong>40 °</strong>" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. It’s given that triangle <strong>X Y Z</strong> is similar to triangle <strong>R S T</strong>, such that <strong>X</strong>, <strong>Y</strong>, and <strong>Z</strong> correspond to <strong>R</strong>, <strong>S</strong>, and <strong>T</strong>, respectively. Since corresponding angles of similar triangles are congruent, it follows that the measure of <strong>angle Z</strong> is congruent to the measure of <strong>angle T</strong>. It’s given that the measure of <strong>angle Z</strong> is <strong>20 °</strong>. Therefore, the measure of <strong>angle T</strong> is <strong>20 °</strong>.<br>Choice A is incorrect and may result from a conceptual error.<br>Choice B is incorrect. This is half the measure of <strong>angle Z</strong>.<br>Choice D is incorrect. This is twice the measure of <strong>angle Z</strong>.",
    hasFigure: false,
  },
  {
    id: "5b4757df",
    type: "spr",
    questionHtml:
      "In triangle <strong>R S T</strong>, angle <strong>T</strong> is a right angle, point <strong>L</strong> lies on <strong>R S</strong>, point <strong>K</strong> lies on <strong>S T</strong>, and <strong>L K</strong> is parallel to <strong>R T</strong>. If the length of <strong>R T</strong> is <strong>72</strong> units, the length of <strong>L K</strong> is <strong>24</strong> units, and the area of triangle <strong>R S T</strong> is <strong>792</strong> square units, what is the length of <strong>K T</strong>, in units?",
    choices: [],
    correctAnswer: "14.66, 14.67, 44/3",
    explanation:
      "The correct answer is <strong>(44) / (3)</strong>. It's given that in triangle <strong>R S T</strong>, angle <strong>T</strong> is a right angle. The area of a right triangle can be found using the formula <strong>A = one half script l 1 script l 2</strong>, where <strong>A</strong> represents the area of the right triangle, <strong>script l 1</strong> represents the length of one leg of the triangle, and <strong>script l 2</strong> represents the length of the other leg of the triangle. In triangle <strong>R S T</strong>, the two legs are <strong>R T</strong> and <strong>S T</strong>. Therefore, if the length of <strong>R T</strong> is <strong>72</strong> and the area of triangle <strong>R S T</strong> is <strong>792</strong>, then <strong>792 = one half (72) (S T)</strong>, or <strong>792 = (36) (S T)</strong>. Dividing both sides of this equation by <strong>36</strong> yields <strong>22 = S T</strong>. Therefore, the length of <strong>S T</strong> is <strong>22</strong>. It's also given that point <strong>L</strong> lies on <strong>R S</strong>, point <strong>K</strong> lies on <strong>S T</strong>, and <strong>L K</strong> is parallel to <strong>R T</strong>. It follows that angle <strong>L K S</strong> is a right angle. Since triangles <strong>R S T</strong> and <strong>L S K</strong> share angle <strong>S</strong> and have right angles <strong>T</strong> and <strong>K</strong>, respectively, triangles <strong>R S T</strong> and <strong>L S K</strong> are similar triangles. Therefore, the ratio of the length of <strong>R T</strong> to the length of <strong>L K</strong> is equal to the ratio of the length of <strong>S T</strong> to the length of <strong>S K</strong>. If the length of <strong>R T</strong> is <strong>72</strong> and the length of <strong>L K</strong> is <strong>24</strong>, it follows that the ratio of the length of <strong>R T</strong> to the length of <strong>L K</strong> is <strong>(72) / (24)</strong>, or <strong>3</strong>, so the ratio of the length of <strong>S T</strong> to the length of <strong>S K</strong> is <strong>3</strong>. Therefore, <strong>(22) / (S K) = 3</strong>. Multiplying both sides of this equation by <strong>S K</strong> yields <strong>22 = (3) (S K)</strong>. Dividing both sides of this equation by <strong>3</strong> yields <strong>(22) / (3) = S K</strong>. Since the length of <strong>S T</strong>, <strong>22</strong>, is the sum of the length of <strong>S K</strong>, <strong>(22) / (3)</strong>, and the length of <strong>K T</strong>, it follows that the length of <strong>K T</strong> is <strong>22 − (22) / (3)</strong>, or <strong>(44) / (3)</strong>. Note that 44/3, 14.66, and 14.67 are examples of ways to enter a correct answer.",
    hasFigure: false,
  },
  {
    id: "6a3fbec3",
    type: "spr",
    questionHtml:
      "In the figure above, <br> <strong>the length of B D = 6</strong> and <br> <strong>the length of A D = 8</strong>. What is the length of <br> <strong>D C</strong> ?",
    choices: [],
    correctAnswer: "",
    explanation:
      "The correct answer is 4.5. According to the properties of right triangles, BD divides triangle ABC into two similar triangles, ABD and BCD. The corresponding sides of ABD and BCD are proportional, so the ratio of BD to AD is the same as the ratio of DC to BD. Expressing this information as a proportion gives <strong>six eighths = the fraction D C, over 6</strong>. Solving the proportion for DC results in <br> <strong>D C = 4 . 5</strong>. Note that 4.5 and 9/2 are examples of ways to enter a correct answer.",
    hasFigure: true,
    figureUrl: "/practice-images/6a3fbec3_img1.png",
  },
  {
    id: "6d99b141",
    type: "spr",
    questionHtml:
      "Triangle upper A upper C upper D partially overlaps triangle upper E upper B upper D.<br>Point upper C is on line segment upper B upper D.<br>Point upper E is on line segment upper A upper D.<br>The measure of angle upper A upper E upper B is x°.<br>A note indicates the figure is not drawn to scale.<br><br> <br>In the figure, <strong>A C = C D</strong>. The measure of angle <strong>E B C</strong> is <strong>45 °</strong>, and the measure of angle <strong>A C D</strong> is <strong>104 °</strong>. What is the value of <strong>x</strong>?",
    choices: [],
    correctAnswer: "83",
    explanation:
      "The correct answer is <strong>83</strong>. It's given that in the figure, <strong>A C = C D</strong>. Thus, triangle <strong>A C D</strong> is an isosceles triangle and the measure of angle <strong>C D A</strong> is equal to the measure of angle <strong>C A D</strong>. The sum of the measures of the interior angles of a triangle is <strong>180 °</strong>. Thus, the sum of the measures of the interior angles of triangle <strong>A C D</strong> is <strong>180 °</strong>. It's given that the measure of angle <strong>A C D</strong> is <strong>104 °</strong>. It follows that the sum of the measures of angles <strong>C D A</strong> and <strong>C A D</strong> is <strong>(180 − 104) °</strong>, or <strong>76 °</strong>. Since the measure of angle <strong>C D A</strong> is equal to the measure of angle <strong>C A D</strong>, the measure of angle <strong>C D A</strong> is half of <strong>76 °</strong>, or <strong>38 °</strong>. The sum of the measures of the interior angles of triangle <strong>B D E</strong> is <strong>180 °</strong>. It's given that the measure of angle <strong>E B C</strong> is <strong>45 °</strong>. Since the measure of angle <strong>B D E</strong>, which is the same angle as angle <strong>C D A</strong>, is <strong>38 °</strong>, it follows that the measure of angle <strong>D E B</strong> is <strong>(180 − 45 − 38) °</strong>, or <strong>97 °</strong>. Since angle <strong>D E B</strong> and angle <strong>A E B</strong> form a straight line, the sum of the measures of these angles is <strong>180 °</strong>. It's given in the figure that the measure of angle <strong>A E B</strong> is <strong>x °</strong>. It follows that <strong>97 + x = 180</strong>. Subtracting <strong>97</strong> from both sides of this equation yields <strong>x = 83</strong>.",
    hasFigure: true,
    figureUrl: "/practice-images/6d99b141_svg1.svg",
  },
  {
    id: "740bf79f",
    type: "mcq",
    questionHtml:
      "In the figure above, what is the length of <strong>NQ</strong> ?",
    choices: [
      { label: "A", text: "2.2" },
      { label: "B", text: "2.3" },
      { label: "C", text: "2.4" },
      { label: "D", text: "2.5" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. First, <strong>M P</strong> is the hypotenuse of right <strong>triangle M N P</strong>, whose legs have lengths 3 and 4. Therefore, <strong>(the length of M P, ), ² = 3² + 4²</strong>, so <strong>(the length of M P, ), ² = 25</strong> and <strong>the length of M P = 5</strong>. Second, because <strong>angle M N P</strong> corresponds to <strong>angle N Q P</strong> and because <strong>angle M P N</strong> corresponds to <strong>angle N P Q</strong>, <strong>triangle M N P</strong> is similar to <strong>triangle N Q P</strong>. The ratio of corresponding sides of similar triangles is constant, so <strong>the length of N Q over the length of M N = the length of N P over the length of M P</strong>. Since <strong>M P = 5</strong> and it’s given that <strong>the length of M N = 3</strong> and <strong>the length of N P = 4</strong>, <strong>the length of N Q, over 3 = 4 over 5</strong>. Solving for NQ results in <strong>the length of N Q = 12 over 5</strong>, or 2.4.Choices A, B, and D are incorrect and may result from setting up incorrect ratios.",
    hasFigure: true,
    figureUrl: "/practice-images/740bf79f_img1.png",
  },
  {
    id: "901c3215",
    type: "mcq",
    questionHtml:
      "In triangles <strong>A B C</strong> and <strong>D E F</strong>, angles <strong>B</strong> and <strong>E</strong> each have measure <strong>27 °</strong> and angles <strong>C</strong> and <strong>F</strong> each have measure <strong>41 °</strong>. Which additional piece of information is sufficient to determine whether triangle <strong>A B C</strong> is congruent to triangle <strong>D E F</strong>?",
    choices: [
      { label: "A", text: "The measure of angle <strong>A</strong>" },
      { label: "B", text: "The length of side <strong>A B</strong>" },
      {
        label: "C",
        text: "The lengths of sides <strong>B C</strong> and <strong>E F</strong>",
      },
      { label: "D", text: "No additional information is necessary." },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. Since angles <strong>B</strong> and <strong>E</strong> each have the same measure and angles <strong>C</strong> and <strong>F</strong> each have the same measure, triangles <strong>A B C</strong> and <strong>D E F</strong> are similar, where side <strong>B C</strong> corresponds to side <strong>E F</strong>. To determine whether two similar triangles are congruent, it is sufficient to determine whether one pair of corresponding sides are congruent. Therefore, to determine whether triangles <strong>A B C</strong> and <strong>D E F</strong> are congruent, it is sufficient to determine whether sides <strong>B C</strong> and <strong>E F</strong> have equal length. Thus, the lengths of <strong>B C</strong> and <strong>E F</strong> are sufficient to determine whether triangle <strong>A B C</strong> is congruent to triangle <strong>D E F</strong>.<br>Choice A is incorrect and may result from conceptual errors.<br>Choice B is incorrect and may result from conceptual errors.<br>Choice D is incorrect. The given information is sufficient to determine that triangles <strong>A B C</strong> and <strong>D E F</strong> are similar, but not whether they are congruent.",
    hasFigure: false,
  },
  {
    id: "947a3cde",
    type: "spr",
    questionHtml:
      "In the figure above, <strong>M Q</strong> and <strong>N R</strong> intersect at point P, <strong>N P = Q P</strong>, and <strong>M P = P R</strong>. What is the measure, in degrees, of <strong>angle Q M R</strong> ? (Disregard the degree symbol when gridding your answer.)",
    choices: [],
    correctAnswer: "",
    explanation:
      "The correct answer is 30. It is given that the measure of <strong>angle Q P R</strong> is <strong>60 °</strong>. Angle MPR and <strong>angle Q P R</strong> are collinear and therefore are supplementary angles. This means that the sum of the two angle measures is <strong>180 °</strong>, and so the measure of <strong>angle M P R</strong> is <strong>120 °</strong>. The sum of the angles in a triangle is <strong>180 °</strong>. Subtracting the measure of <strong>angle M P R</strong> from <strong>180 °</strong> yields the sum of the other angles in the triangle MPR. Since <strong>180 − 120 = 60</strong>, the sum of the measures of <strong>angle Q M R</strong> and <strong>angle N R M</strong> is <strong>60 °</strong>. It is given that <strong>the length of side M P = the length of side P R</strong>, so it follows that triangle MPR is isosceles. Therefore<strong>angle Q M R</strong> and <strong>angle N R M</strong> must be congruent. Since the sum of the measure of these two angles is <strong>60 °</strong>, it follows that the measure of each angle is <strong>30 °</strong>.An alternate approach would be to use the exterior angle theorem, noting that the measure of <strong>angle Q P R</strong> is equal to the sum of the measures of <strong>angle Q M R</strong> and <strong>angle N R M</strong>. Since both angles are equal, each of them has a measure of <strong>30 °</strong>.",
    hasFigure: true,
    figureUrl: "/practice-images/947a3cde_img1.png",
  },
  {
    id: "a0369739",
    type: "mcq",
    questionHtml:
      "In triangle <strong>A B C</strong>, the measure of angle <strong>B</strong> is <strong>90 °</strong> and <strong>B D</strong> is an altitude of the triangle. The length of <strong>A B</strong> is <strong>15</strong> and the length of <strong>A C</strong> is <strong>23</strong> greater than the length of <strong>A B</strong>. What is the value of <strong>(B C) / (B D)</strong>?",
    choices: [
      { label: "A", text: "<strong>(15) / (38)</strong>" },
      { label: "B", text: "<strong>(15) / (23)</strong>" },
      { label: "C", text: "<strong>(23) / (15)</strong>" },
      { label: "D", text: "<strong>(38) / (15)</strong>" },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. It's given that in triangle <strong>A B C</strong>, the measure of angle <strong>B</strong> is <strong>90 °</strong> and  <strong>B D</strong> is an altitude of the triangle. Therefore, the measure of angle <strong>B D C</strong> is <strong>90 °</strong>. It follows that angle <strong>B</strong> is congruent to angle <strong>D</strong> and angle <strong>C</strong> is congruent to angle <strong>C</strong>. By the angle-angle similarity postulate, triangle <strong>A B C</strong> is similar to triangle <strong>B D C</strong>. Since triangles <strong>A B C</strong> and <strong>B D C</strong> are similar, it follows that <strong>(A C) / (A B) = (B C) / (B D)</strong>. It's also given that the length of <strong>A B</strong> is <strong>15</strong> and the length of <strong>A C</strong> is <strong>23</strong> greater than the length of <strong>A B</strong>. Therefore, the length of <strong>A C</strong> is <strong>15 + 23</strong>, or <strong>38</strong>. Substituting <strong>15</strong> for <strong>A B</strong> and <strong>38</strong> for <strong>A C</strong> in the equation <strong>(A C) / (A B) = (B C) / (B D)</strong> yields <strong>(38) / (15) = (B C) / (B D)</strong>. Therefore, the value of <strong>(B C) / (B D)</strong> is <strong>(38) / (15)</strong>.<br>Choice A is incorrect. This is the value of <strong>(B D) / (B C)</strong>.<br>Choice B is incorrect and may result from conceptual or calculation errors.<br>Choice C is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "bd7f6e30",
    type: "mcq",
    questionHtml:
      "In the figure above, <strong>R T = T U</strong>. What is the value of x ?",
    choices: [
      { label: "A", text: "72" },
      { label: "B", text: "66" },
      { label: "C", text: "64" },
      { label: "D", text: "58" },
    ],
    correctAnswer: "C",
    explanation:
      "Choice C is correct. Since <strong>the length of side R T = the length of side T U</strong>, it follows that <strong>triangle R T U</strong> is an isosceles triangle with base RU. Therefore, <strong>angle T R U</strong> and <strong>angle T U R</strong> are the base angles of an isosceles triangle and are congruent. Let the measures of both <strong>angle T R U</strong> and <strong>angle T U R</strong> be <strong>t °</strong>. According to the triangle sum theorem, the sum of the measures of the three angles of a triangle is <strong>180 °</strong>. Therefore, <strong>114 ° + 2 t ° = 180 °</strong>, so <strong>t = 33</strong>.Note that <strong>angle T U R</strong> is the same angle as <strong>angle S U V</strong>. Thus, the measure of <strong>angle S U V</strong> is <strong>33 °</strong>. According to the triangle exterior angle theorem, an external angle of a triangle is equal to the sum of the opposite interior angles. Therefore, <strong>x °</strong> is equal to the sum of the measures of <strong>angle V S U</strong> and <strong>angle S U V</strong>; that is, <strong>31 ° + 33 ° = 64 °</strong>. Thus, the value of x is 64.<br>Choice B is incorrect. This is the measure of <strong>angle S T R</strong>, but <strong>angle S T R</strong> is not congruent to <strong>angle S V R</strong>. Choices A and D are incorrect and may result from a calculation error.",
    hasFigure: true,
    figureUrl: "/practice-images/bd7f6e30_img1.png",
  },
  {
    id: "e10d8313",
    type: "spr",
    questionHtml:
      "From left to right, horizontal line segment upper P upper V includes the following points:<br><br>Upper P<br>Upper Q<br>Upper R<br>Upper S<br>Upper T<br>Upper V<br><br>Points upper X and upper U are each below line segment upper P upper V.<br>From left to right, the following 2 overlapping triangles are formed between points on line segment upper P upper V and points upper X and upper U:<br><br>Triangle upper Q upper X upper S<br>Triangle upper R upper U upper T<br><br>Line segments upper R upper U and upper S upper X intersect at point upper W.<br>A note indicates the figure is not drawn to scale.<br><br>In the figure shown, points <strong>Q</strong>, <strong>R</strong>, <strong>S</strong>, and <strong>T</strong> lie on line segment <strong>P V</strong>, and line segment <strong>R U</strong> intersects line segment <strong>S X</strong> at point <strong>W</strong>. The measure of <strong>angle S Q X</strong> is <strong>48 °</strong>, the measure of <strong>angle S X Q</strong> is <strong>86 °</strong>, the measure of <strong>angle S W U</strong> is <strong>85 °</strong>, and the measure of <strong>angle V T U</strong> is <strong>162 °</strong>. What is the measure, in degrees, of <strong>angle T U R</strong>?",
    choices: [],
    correctAnswer: "123",
    explanation:
      "The correct answer is <strong>123</strong>. The triangle angle sum theorem states that the sum of the measures of the interior angles of a triangle is <strong>180</strong> degrees. It's given that the measure of <strong>angle S Q X</strong> is <strong>48 °</strong> and the measure of <strong>angle S X Q</strong> is <strong>86 °</strong>. Since points <strong>S</strong>, <strong>Q</strong>, and <strong>X</strong> form a triangle, it follows from the triangle angle sum theorem that the measure, in degrees, of <strong>angle Q S X</strong> is <strong>180 − 48 − 86</strong>, or <strong>46</strong>. It's also given that the measure of <strong>angle S W U</strong> is <strong>85 °</strong>. Since <strong>angle S W U</strong> and <strong>angle S W R</strong> are supplementary angles, the sum of their measures is <strong>180</strong> degrees. It follows that the measure, in degrees, of <strong>angle S W R</strong> is <strong>180 − 85</strong>, or <strong>95</strong>. Since points <strong>R</strong>, <strong>S</strong>, and <strong>W</strong> form a triangle, and <strong>angle R S W</strong> is the same angle as <strong>angle Q S X</strong>, it follows from the triangle angle sum theorem that the measure, in degrees, of <strong>angle W R S</strong> is <strong>180 − 46 − 95</strong>, or <strong>39</strong>. It's given that the measure of <strong>angle V T U</strong> is <strong>162 °</strong>. Since <strong>angle V T U</strong> and <strong>angle S T U</strong> are supplementary angles, the sum of their measures is <strong>180</strong> degrees. It follows that the measure, in degrees, of <strong>angle S T U</strong> is <strong>180 − 162</strong>, or <strong>18</strong>. Since points <strong>R</strong>, <strong>T</strong>, and <strong>U</strong> form a triangle, and <strong>angle U R T</strong> is the same angle as <strong>angle W R S</strong>, it follows from the triangle angle sum theorem that the measure, in degrees, of <strong>angle T U R</strong> is <strong>180 − 39 − 18</strong>, or <strong>123</strong>.",
    hasFigure: true,
    figureUrl: "/practice-images/e10d8313_svg1.svg",
  },
  {
    id: "eeb4143c",
    type: "spr",
    questionHtml:
      "The area of triangle ABC above is at least 48 but no more than 60. If y is an integer, what is one possible value of x ?",
    choices: [],
    correctAnswer: "",
    explanation:
      "The correct answer is either <strong>the fraction 10 over 3</strong>, <strong>the fraction 15 over 4</strong>, or <strong>the fraction 25 over 6</strong>. The area of triangle ABC can be expressed as <strong>one half · (5 + 7, ) · y</strong> or <strong>6 y</strong>. It’s given that the area of triangle ABC is at least 48 but no more than 60. It follows that <strong>48 ≤ 6 y, which ≤ 60</strong>. Dividing by 6 to isolate y in this compound inequality yields <strong>8 ≤ y, which ≤ 10</strong>. Since y is an integer, <strong>y = 8, 9, or 10</strong>. In the given figure, the two right triangles shown are similar because they have two pairs of congruent angles: their respective right angles and angle A. Therefore, the following proportion is true: <strong>the fraction x over y = the fraction 5 over 12</strong>. Substituting 8 for y in the proportion results in <strong>the fraction x over 8 = the fraction 5 over 12</strong>. Cross multiplying and solving for x yields <strong>the fraction 10 over 3</strong>. Substituting 9 for y in the proportion results in <strong>the fraction x over 9 = the fraction 5 over 12</strong>. Cross multiplying and solving for x yields <strong>the fraction 15 over 4</strong>. Substituting 10 for y in the proportion results in <strong>the fraction x over 10 = the fraction 5 over 12</strong>. Cross multiplying and solving for x yields <strong>the fraction 25 over 6</strong>. Note that 10/3, 15/4, 25/6, 3.333, 3.75, 4.166, and 4.167 are examples of ways to enter a correct answer.",
    hasFigure: true,
    figureUrl: "/practice-images/eeb4143c_img1.png",
  },
  {
    id: "f7dbde16",
    type: "mcq",
    questionHtml:
      "In triangles <strong>L M N</strong> and <strong>R S T</strong>, angles <strong>L</strong> and <strong>R</strong> each have measure <strong>60 °</strong>, <strong>L N = 10</strong>, and <strong>R T = 30</strong>. Which additional piece of information is sufficient to prove that triangle <strong>L M N</strong> is similar to triangle <strong>R S T</strong>?",
    choices: [
      {
        label: "A",
        text: "<strong>M N = 7</strong> and <strong>S T = 7</strong>",
      },
      {
        label: "B",
        text: "<strong>M N = 7</strong> and <strong>S T = 21</strong>",
      },
      {
        label: "C",
        text: "The measures of angles <strong>M</strong> and <strong>S</strong> are <strong>70 °</strong> and <strong>60 °</strong>, respectively.",
      },
      {
        label: "D",
        text: "The measures of angles <strong>M</strong> and <strong>T</strong> are <strong>70 °</strong> and <strong>50 °</strong>, respectively.",
      },
    ],
    correctAnswer: "D",
    explanation:
      "Choice D is correct. Two triangles are similar if they have three pairs of congruent corresponding angles. It’s given that angles <strong>L</strong> and <strong>R</strong> each measure <strong>60 °</strong>, and so these corresponding angles are congruent. If angle <strong>M</strong> is <strong>70 °</strong>, then angle <strong>N</strong> must be <strong>50 °</strong> so that the sum of the angles in triangle <strong>L M N</strong> is <strong>180 °</strong>. If angle <strong>T</strong> is <strong>50 °</strong>, then angle <strong>S</strong> must be <strong>70 °</strong> so that the sum of the angles in triangle <strong>R S T</strong> is <strong>180 °</strong>. Therefore, if the measures of angles <strong>M</strong> and <strong>T</strong> are <strong>70 °</strong> and <strong>50 °</strong>, respectively, then corresponding angles <strong>M</strong> and <strong>S</strong> are both <strong>70 °</strong>, and corresponding angles <strong>N</strong> and <strong>T</strong> are both <strong>50 °</strong>. It follows that triangles <strong>L M N</strong> and <strong>R S T</strong> have three pairs of congruent corresponding angles, and so the triangles are similar. Therefore, the additional piece of information that is sufficient to prove that triangle <strong>L M N</strong> is similar to triangle <strong>R S T</strong> is that the measures of angles <strong>M</strong> and <strong>T</strong> are <strong>70 °</strong> and <strong>50 °</strong>, respectively.<br>Choice A is incorrect. If the measures of two sides in one triangle are proportional to the corresponding sides in another triangle and the included angles are congruent, then the triangles are similar. However, the two sides given are not proportional and the angle given is not included by the given sides.<br>Choice B is incorrect. If the measures of two sides in one triangle are proportional to the corresponding sides in another triangle and the included angles are congruent, then the triangles are similar. However, the angle given is not included between the proportional sides.<br>Choice C is incorrect and may result from conceptual or calculation errors.",
    hasFigure: false,
  },
  {
    id: "f88f27e5",
    type: "spr",
    questionHtml: "What is the value of x ?",
    choices: [],
    correctAnswer: "",
    explanation:
      "The correct answer is 97. The intersecting lines form a triangle, and the angle with measure of <strong>x °</strong> is an exterior angle of this triangle. The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles of the triangle. One of these angles has measure of <strong>23 °</strong> and the other, which is supplementary to the angle with measure <strong>106 °</strong>, has measure of <strong>180 ° − 106 ° = 74 °</strong>. Therefore, the value of x is <strong>23 + 74 = 97</strong>.",
    hasFigure: true,
    figureUrl: "/practice-images/f88f27e5_img1.png",
  },
];
