📐 SAT / AMC Style · 20 Questions

Geometry Master Quiz

All Core Topics · Multiple Choice · Timed · Detailed Solutions

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Core Concepts
Angles

Angle Relationships

Supplementary: a + b = 180° Complementary: a + b = 90° Vertical angles: a = b

Angles on a straight line sum to 180°. Vertical angles (opposite) are equal.

Triangles

Triangle Properties

Angle sum: A + B + C = 180° Exterior angle = sum of two non-adjacent interior angles Isosceles: two equal sides → two equal base angles

The exterior angle theorem is heavily tested.

Pythagorean Theorem

Right Triangles

a² + b² = c² (c = hypotenuse) Common sets: 3-4-5, 5-12-13, 8-15-17, 7-24-25

Memorize the common Pythagorean triples!

Circles

Circle Formulas

Circumference = 2πr = πd Area = πr² Arc length = (θ/360°) × 2πr

r = radius, d = diameter. Area uses r², circumference uses r.

Coordinate Geometry

Points & Lines

Midpoint: ((x₁+x₂)/2, (y₁+y₂)/2) Distance: √((x₂-x₁)²+(y₂-y₁)²) Slope: m = (y₂-y₁)/(x₂-x₁)

Distance formula is Pythagorean theorem in disguise.

Area & Volume

Key Formulas

Triangle: ½ × base × height Trapezoid: ½(b₁+b₂) × h Cylinder: V = πr²h Rect. prism: V = l × w × h Cube surface area: 6s²

Trapezoid uses the average of both bases × height.

Parallel Lines

Transversal Angles

Alternate interior angles: equal Corresponding angles: equal Co-interior (same-side): sum = 180°

When two parallel lines are cut by a transversal, alternate interior angles are equal.

Similarity & Congruence

Triangle Tests

Congruence: SSS, SAS, ASA, AAS, HL Similarity: AA, SAS~, SSS~ Ratio k → Area ratio k²

Similar triangles have proportional sides. If sides ratio is k, perimeter ratio is k too.

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Must-Memorize Checklist

  • Triangle angles sum = 180°; Quadrilateral angles sum = 360°
  • Pythagorean triples: 3-4-5, 5-12-13, 8-15-17
  • Circle: C = 2πr, A = πr²
  • Exterior angle of triangle = sum of two non-adjacent interior angles
  • Parallel lines: alternate interior angles are equal; co-interior angles sum to 180°
  • Congruence criteria: SSS, SAS, ASA, AAS, HL (right triangle only)
  • Midpoint: average the x-coords, average the y-coords
  • Distance = √(Δx² + Δy²)
  • Volume of cylinder = πr²h; Cube surface area = 6s²
  • Isosceles triangle: two equal sides → two equal base angles
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Worked Example
EXAMPLE

In triangle ABC, angle A = 48° and angle B = 67°. What is the measure of angle C?

Solution: The sum of angles in any triangle is 180°.
∴ C = 180° − 48° − 67° = 65°
EXAMPLE

A right triangle has legs of length 5 and 12. Find the hypotenuse.

Solution: By the Pythagorean theorem: c² = 5² + 12² = 25 + 144 = 169
∴ c = √169 = 13  (recognise the 5-12-13 triple!)