📐 Exam Preparation Series

Geometry
Master Quiz

20 Essential Problems — All Major Topics

20Questions
40Minutes
10Topics


Angles & Lines Triangles Pythagorean Theorem Circles Area & Perimeter Coordinate Geometry Transformations 3D Solids Similar Figures Proofs & Logic

📘 Key Concepts & Formulas

Study these before you attempt the quiz

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Angles & Parallel Lines

  • Complementary angles sum to 90°
  • Supplementary angles sum to 180°
  • Vertical angles are equal
  • Parallel lines cut by transversal: alternate interior angles are equal; co-interior (same-side) angles sum to 180°
  • Angles in a triangle sum to 180°
  • Exterior angle of triangle = sum of two non-adjacent interior angles
Exterior Angle Theorem Exterior angle = α + β (remote interior angles)
Example A triangle has angles 55° and 70°. Find the third angle.
180 − 55 − 70 = 55°
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Triangles & Pythagorean Theorem

  • Right triangle: a² + b² = c² (c = hypotenuse)
  • Special triangles: 3-4-5, 5-12-13, 8-15-17
  • 45-45-90: legs = x, hypotenuse = x√2
  • 30-60-90: short = x, long = x√3, hyp = 2x
  • Isosceles triangle: 2 equal sides, 2 equal base angles
  • Equilateral: all sides equal, all angles = 60°
Pythagorean Theorem a² + b² = c²
Area of Triangle A = ½ × base × height
Example Legs = 6 and 8. Find hypotenuse.
c = √(36 + 64) = √100 = 10
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Circles

  • Circumference = 2πr = πd
  • Area = πr²
  • Arc length = (θ/360) × 2πr
  • Sector area = (θ/360) × πr²
  • Central angle = intercepted arc
  • Inscribed angle = ½ × intercepted arc
  • Tangent ⊥ radius at point of tangency
Key Formulas C = 2πr   |   A = πr²   |   Inscribed angle = arc / 2
Example Circle radius = 5. Find area.
A = π(5²) = 25π ≈ 78.54
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Quadrilaterals & Polygons

  • Sum of interior angles of n-gon = (n − 2) × 180°
  • Rectangle: A = l × w, diagonals equal & bisect each other
  • Parallelogram: opposite sides parallel & equal, opposite angles equal
  • Rhombus: all sides equal, diagonals perpendicular bisectors
  • Trapezoid: A = ½(b₁ + b₂) × h
Interior Angle Sum S = (n − 2) × 180°
Example Find the sum of interior angles of a hexagon.
S = (6 − 2) × 180 = 720°
xy

Coordinate Geometry

  • Distance = √[(x₂−x₁)² + (y₂−y₁)²]
  • Midpoint = ((x₁+x₂)/2, (y₁+y₂)/2)
  • Slope m = (y₂−y₁)/(x₂−x₁)
  • Parallel lines: equal slopes (m₁ = m₂)
  • Perpendicular lines: slopes multiply to −1 (m₁ × m₂ = −1)
  • Equation of circle: (x−h)² + (y−k)² = r²
Slope-Intercept Form y = mx + b   |   m = rise/run
Example Find distance between (1, 2) and (4, 6).
d = √(9 + 16) = √25 = 5
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Similar & Congruent Figures

  • Similar figures: same shape, proportional sides, equal angles
  • Scale factor k → area ratio k², volume ratio k³
  • Triangle similarity: AA, SAS, SSS
  • Triangle congruence: SSS, SAS, ASA, AAS, HL
  • CPCTC: Corresponding Parts of Congruent Triangles are Congruent
Scale Factor Rule Sides ratio = k   |   Area ratio = k²   |   Volume ratio = k³
3D

3D Solids

  • Cube: V = s³, SA = 6s²
  • Rectangular prism: V = lwh, SA = 2(lw + lh + wh)
  • Cylinder: V = πr²h, SA = 2πr² + 2πrh
  • Cone: V = ⅓πr²h, SA = πr² + πrl (l = slant height)
  • Sphere: V = ⁴⁄₃πr³, SA = 4πr²
  • Pyramid: V = ⅓ × base area × height
Volume Quick Reference Prism/Cylinder: V = Bh   |   Cone/Pyramid: V = ⅓Bh
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📋 Results & Full Solutions

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