import React, { useRef, useState, useEffect } from "react"; import { ArrowDown, Check, Target, Layers, Calculator, BookOpen, } from "lucide-react"; import InteractiveTransversal from "../../../components/lessons/InteractiveTransversal"; import InteractiveTriangle from "../../../components/lessons/InteractiveTriangle"; import PolygonWidget from "../../../components/lessons/PolygonWidget"; import Quiz from "../../../components/lessons/Quiz"; import { ANGLES_QUIZ_DATA } from "../../../utils/constants"; import { Frac } from "../../../components/Math"; interface LessonProps { onFinish?: () => void; } const LinesAnglesLesson: React.FC = ({ onFinish }) => { const [activeSection, setActiveSection] = useState(0); const sectionsRef = useRef<(HTMLElement | null)[]>([]); const scrollToSection = (index: number) => { setActiveSection(index); sectionsRef.current[index]?.scrollIntoView({ behavior: "smooth", block: "start", }); }; useEffect(() => { const observer = new IntersectionObserver( (entries) => { entries.forEach((entry) => { if (entry.isIntersecting) { const index = sectionsRef.current.indexOf( entry.target as HTMLElement, ); if (index !== -1) setActiveSection(index); } }); }, { rootMargin: "-20% 0px -60% 0px" }, ); sectionsRef.current.forEach((section) => { if (section) observer.observe(section); }); return () => observer.disconnect(); }, []); const SectionMarker = ({ index, title, icon: Icon, }: { index: number; title: string; icon: any; }) => { const isActive = activeSection === index; const isPast = activeSection > index; return ( ); }; return (
{/* Section 1: Parallel Lines */}
{ sectionsRef.current[0] = el; }} className="min-h-screen flex flex-col justify-center mb-24 pt-20 lg:pt-0" >

Parallel Lines & Transversals

When two parallel lines are cut by a transversal, 8 angles are formed. They fall into exactly two groups:{" "} equal angles and{" "} supplementary pairs (summing to 180°). Know one angle → find all eight.

The 5 Angle-Pair Relationships

Angle Pair Location Relationship
Corresponding Same side, same position at each intersection Equal
Alternate Interior Between the parallel lines, opposite sides Equal
Alternate Exterior Outside the parallel lines, opposite sides Equal
Co-Interior (Same-Side) Between the lines, same side of transversal Supplementary (sum = 180°)
Vertical Angles Opposite each other at an intersection Equal
{/* Worked Example */}

Worked Example: Find All 8 Angles

If one angle formed by a transversal cutting two parallel lines is 65°, find all other angles.

Angle 1 = 65° (given)

Vertical angle = 65° (vertical angles are equal)

Corresponding angle = 65° (corresponding angles are equal)

Its vertical angle = 65°

All four supplementary angles = 180° − 65° ={" "} 115°

Result: four 65° angles and four 115° angles.

SAT Strategy: Label with x and 180 − x

When angles are expressed algebraically, label all equal angles as "x" and all supplementary angles as "180 − x." Then set equal or add to 180 to solve.

Example: Corresponding angles → 3x + 15 = 2x + 45 → x = 30°

{/* Section 2: Triangles */}
{ sectionsRef.current[1] = el; }} className="min-h-screen flex flex-col justify-center mb-24" >

Triangle Theorems

Two essential theorems unlock almost every SAT triangle problem. The interactive tool below lets you drag vertices to verify both dynamically.

Core Triangle Rules

Triangle Sum Theorem

∠A + ∠B + ∠C = 180°

The three interior angles of any triangle always sum to exactly 180°. No exceptions.

Exterior Angle Theorem

∠ext = ∠A + ∠B

An exterior angle equals the sum of the two non-adjacent (remote) interior angles.

{/* Worked Examples */}

Worked Examples

Example 1: Find a missing angle

Two angles are 47° and 83°. Find the third.

∠C = 180° − 47° − 83° ={" "} 50°

Example 2: Exterior angle

Two interior angles are 40° and 65°. Find the exterior angle at the third vertex.

∠ext = 40° + 65° ={" "} 105°

Example 3: Isosceles triangle

An isosceles triangle has vertex angle = 40°. Find the base angles.

Each base angle = ={" "} ={" "} 70°

{/* Triangle Inequality */}

Triangle Inequality Theorem

Any two sides of a triangle must sum to more than the third side.

a + b > c, a + c > b, b + c > a

Example: Can a triangle have sides 3, 5, 9?

3 + 5 = 8 < 9 → NO, not a valid triangle.

{/* Section 3: Special Triangles */}
{ sectionsRef.current[2] = el; }} className="min-h-screen flex flex-col justify-center mb-24" >

Special Right Triangles

These two triangle types appear constantly on the SAT. Memorize their side ratios so you can find any missing side without using the Pythagorean theorem.

The Two Special Right Triangles

45° — 45° — 90°

Sides: 1 : 1 : √2

Two equal legs. Hypotenuse = leg × √2.

If leg = 5: hyp = 5√2

If hyp = 8: leg = = 4√2

30° — 60° — 90°

Sides: 1 : √3 : 2

Shortest leg opposite 30°. Hypotenuse = 2 × short leg.

Short leg = 4: long leg = 4√3, hyp = 8

Hyp = 10: short leg = 5, long leg = 5√3

{/* Pythagorean Theorem */}

Pythagorean Theorem & Common Triples

a² + b² = c²

c is always the hypotenuse (opposite the right angle). Memorize these Pythagorean triples — they appear frequently on the SAT:

Triple (a, b, c) Scaled Version Verify
3, 4, 5 6-8-10, 9-12-15 9 + 16 = 25 ✓
5, 12, 13 10-24-26 25 + 144 = 169 ✓
8, 15, 17 64 + 225 = 289 ✓
7, 24, 25 49 + 576 = 625 ✓
{/* Worked examples */}

Worked Example: 30-60-90

An equilateral triangle has side length 10. Find the height.

The height bisects it into two 30-60-90 triangles.

Short leg (half the base) = 5

Long leg (height) = 5√3 ≈{" "} 8.66

Worked Example: Pythagorean Triple

A right triangle has legs 9 and 12. Find the hypotenuse.

Recognize: 9 and 12 are multiples of 3 and 4 (× 3).

This is a 3-4-5 triple × 3: hypotenuse ={" "} 15

{/* Section 4: Polygons */}
{ sectionsRef.current[3] = el; }} className="min-h-screen flex flex-col justify-center mb-24" >

Polygons

Polygon angle rules extend triangle logic — any polygon can be divided into triangles, which is where the interior angle sum formula comes from.

Polygon Angle Formulas

Interior Angle Sum

(n − 2) × 180°

n = number of sides. Triangle: 180° | Quadrilateral: 360° | Pentagon: 540° | Hexagon: 720°

Exterior Angle Sum

Always = 360°

True for ALL convex polygons, regardless of n. Imagine walking around the polygon — you turn a full circle.

{/* Reference table */}

Quick Reference: Regular Polygons

Polygon Sides (n) Interior Sum Each Interior Angle
Triangle 3 180° 60°
Quadrilateral 4 360° 90°
Pentagon 5 540° 108°
Hexagon 6 720° 120°
Octagon 8 1080° 135°
{/* Formula for one angle */}

Each Interior Angle of a Regular n-gon

Example: Regular hexagon → ={" "} = 120°

Example: Regular octagon → ={" "} = 135°

{/* Section 5: Quiz */}
{ sectionsRef.current[4] = el; }} className="min-h-screen flex flex-col justify-center" >

Practice Time

{ANGLES_QUIZ_DATA.map((quiz, idx) => (
Question {idx + 1}
))}

Topic Mastered!

); }; export default LinesAnglesLesson;