Concepts → Examples → 20-Question Test → Answer Key
Study Guide
Geometry Concepts
Master these before the test!
📐
Unit 1
Lines & Angles
🔑 Key Definitions
• Complementary: two angles sum to 90°
• Supplementary: two angles sum to 180°
• Vertical angles: equal (opposite at intersection)
• Linear pair: adjacent angles summing to 180°
• Interior angle sum: 180°
• Exterior angle = sum of 2 non-adjacent interior angles
• Triangle inequality: each side < sum of other two
• Pythagorean theorem: a² + b² = c² (right △)
📐 Area & Special Triangles
Area = (1/2) × base × height
45-45-90: sides x : x : x√2
30-60-90: sides x : x√3 : 2x
In a right triangle, the legs are 5 and 12. Find the hypotenuse.
c² = 5² + 12² = 25 + 144 = 169
c = √169 = 13
Answer: 13
⬡
Unit 3
Polygons & Quadrilaterals
🔑 Polygon Angle Formulas
Sum of interior angles: (n − 2) × 180°
Each interior angle (regular): (n − 2) × 180° / n
Sum of exterior angles: always 360°
Each exterior angle (regular): 360° / n
📐 Quadrilateral Properties
Parallelogram: opposite sides equal & parallel
→ opposite angles equal, consecutive supplementary
Rectangle: parallelogram + all right angles, diagonals equal
Rhombus: parallelogram + all sides equal, diagonals ⊥
Square: all of rectangle + rhombus properties
Trapezoid: exactly 1 pair of parallel sides (bases)
⭐ Memorize!
Every quadrilateral interior sum = 360°
Parallelogram diagonals bisect each other
Rhombus diagonals are perpendicular bisectors
🦕 Rex's Example
Find the sum of interior angles of a hexagon.
n = 6: (6 − 2) × 180° = 4 × 180° = 720°
Answer: 720°
⭕
Unit 4
Circles
🔑 Circle Formulas
Circumference: C = 2πr = πd
Area: A = πr²
Arc length: L = (θ/360°) × 2πr
Sector area: A = (θ/360°) × πr²
📐 Angle Theorems
Central angle = intercepted arc
Inscribed angle = ½ × intercepted arc
Angle in semicircle = 90° (Thales' theorem)
Tangent-radius: always perpendicular (90°)
Tangent-chord angle = ½ × intercepted arc
⭐ Memorize!
Inscribed angle = ½ central angle (same arc)
Two tangents from external point: equal length
Chord-chord angle = ½ (sum of intercepted arcs)
🦕 Rex's Example
A circle has radius 6. Find the area of a sector with central angle 60°.
A = (60/360) × π × 6² = (1/6) × 36π = 6π
Answer: 6π sq units
📦
Unit 5
Area, Surface Area & Volume
🔑 2D Area Formulas
Square: A = s²
Rectangle: A = lw
Triangle: A = ½bh
Parallelogram: A = bh
Trapezoid: A = ½(b₁ + b₂)h
Circle: A = πr²
📐 3D Surface Area & Volume
Rectangular prism: V = lwh, SA = 2(lw+lh+wh)
Cylinder: V = πr²h, SA = 2πr² + 2πrh
Cone: V = ⅓πr²h, SA = πr² + πrl
Sphere: V = (4/3)πr³, SA = 4πr²
Pyramid: V = ⅓Bh (B = base area)
⭐ Memorize!
Cone & pyramid: ⅓ × (cylinder/prism volume)
Sphere SA = 4 × (area of great circle)
Slant height l = √(r² + h²) for cone
🦕 Rex's Example
Find the volume of a cylinder with r = 4 and h = 9.
V = πr²h = π × 4² × 9 = π × 16 × 9 = 144π
Answer: 144π cubic units
📍
Unit 6
Coordinate Geometry & Similarity
🔑 Coordinate Formulas
Distance: d = √((x₂−x₁)² + (y₂−y₁)²)
Midpoint: M = ((x₁+x₂)/2, (y₁+y₂)/2)
Slope: m = (y₂−y₁)/(x₂−x₁)
Parallel lines: same slope (m₁ = m₂)
Perpendicular lines: m₁ × m₂ = −1
📐 Similarity & Congruence
Similar triangles: AA, SAS~, SSS~
Congruent triangles: SSS, SAS, ASA, AAS, HL
Scale factor k: corresponding sides in ratio k
Area ratio: k² | Volume ratio: k³
⭐ Memorize!
Perpendicular slopes: m₁ × m₂ = −1
Similar △: CPCTC for corresponding parts
Altitude to hypotenuse: geometric mean relation