Geometry Master | Core Concepts & 20 Key Problems

📐

Angles & Parallel Lines

Unit 1

When a transversal crosses two parallel lines:

★ Must Memorize

• Alternate interior angles: EQUAL
• Co-interior (same-side) angles: sum = 180°
• Corresponding angles: EQUAL
• Vertical angles: EQUAL

Quick Example

Two parallel lines cut by a transversal. One co-interior angle = 72°.
Find the other co-interior angle.

180° − 72° = 108°

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Triangle Angles

Unit 2

The three interior angles of any triangle always add up to 180°.

★ Must Memorize

∠A + ∠B + ∠C = 180°

Exterior Angle Theorem:
exterior ∠ = sum of two non-adjacent interior ∠s

Quick Example

A triangle has angles 47° and 63°. Find the third.

180° − 47° − 63° = 70°

≅

Congruent & Similar Triangles

Unit 3

Congruent (≅): same shape AND size. Similar (~): same shape, proportional sides.

★ Congruence Criteria

SSS · SAS · ASA · AAS · HL (right △)

★ Similarity — Ratio Rule

If sides are in ratio k:1,
areas are in ratio k²:1

Quick Example

Two similar triangles with ratio 3:1. If small area = 8 cm², find large area.

8 × 3² = 8 × 9 = 72 cm²

📏

Pythagorean Theorem & Special Triangles

Unit 4

★ Pythagorean Theorem (right △)

a² + b² = c² (c = hypotenuse)

★ 30-60-90 Triangle

sides: 1 : √3 : 2
(short leg : long leg : hypotenuse)

★ 45-45-90 Triangle

sides: 1 : 1 : √2
(leg : leg : hypotenuse)

Quick Example

Right triangle legs = 6 and 8. Hypotenuse?

√(36+64) = √100 = 10

⭕

Circles — Angles & Arcs

Unit 5

★ Must Memorize

Central angle = intercepted arc
Inscribed angle = ½ × intercepted arc
Tangent ⊥ radius at point of tangency
Semicircle inscribed angle = 90°

Quick Example

Inscribed angle = 35°. Find the intercepted arc.

arc = 2 × 35° = 70°

📦

Area, Volume & Surface Area

Unit 6

★ Area Formulas

Triangle: ½ × base × height
Trapezoid: ½(b₁+b₂) × h
Circle: πr²

★ Volume Formulas

Cylinder: πr²h
Cone: ⅓πr²h
Sphere: (4/3)πr³

★ Surface Area

Sphere: 4πr²
Cylinder (total): 2πr² + 2πrh

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Coordinate Geometry

Unit 7

★ Must Memorize

Midpoint: ((x₁+x₂)/2, (y₁+y₂)/2)
Distance: √((x₂−x₁)²+(y₂−y₁)²)
Slope: m = (y₂−y₁)/(x₂−x₁)
Parallel: m₁ = m₂
Perpendicular: m₁ × m₂ = −1

Quick Example

Midpoint of (2,4) and (8,10)?

((2+8)/2, (4+10)/2) = (5, 7)

🔷

Polygons — Interior Angles

Unit 8

★ Sum of Interior Angles

Sum = (n − 2) × 180°
(n = number of sides)

Each interior angle (regular polygon):
= (n−2)×180° ÷ n

Quick Example

Sum of interior angles of a hexagon (n=6)?

(6−2) × 180° = 4 × 180° = 720°