Estimated study time: 45 minutes
Content:
GRE Geometry covers lines, angles, triangles, quadrilaterals, circles, and coordinate geometry. The GRE does not test trigonometry or 3D geometry beyond basic volume/surface area formulas. All diagrams on the GRE should be treated as approximately accurate for the general shape, but NOT necessarily drawn to scale — never assume lengths or angles from visual appearance unless the problem states specific values.
Lines and angles: vertical angles are equal. Supplementary angles sum to 180°; complementary angles sum to 90°. A transversal crossing two parallel lines creates equal alternate interior angles and equal corresponding angles. The sum of all angles in any triangle is 180°.
Triangle properties are heavily tested. The Pythagorean Theorem (a² + b² = c², where c is the hypotenuse) applies only to right triangles. Memorize Pythagorean triples: 3-4-5, 5-12-13, 8-15-17, and multiples thereof. Special right triangles: 45-45-90 (sides in ratio 1:1:√2) and 30-60-90 (sides in ratio 1:√3:2). The triangle inequality: the sum of any two sides must be greater than the third side. The area of a triangle = ½ × base × height. For an equilateral triangle with side s: area = (√3/4)s².
Similar triangles have equal corresponding angles and proportional corresponding sides. If triangles ABC and DEF are similar with ratio k:1, then corresponding sides are in ratio k:1 and areas are in ratio k²:1. The altitude from the right angle in a right triangle to the hypotenuse creates two smaller triangles, each similar to the original and to each other.
Circles: area = πr², circumference = 2πr = πd. Arc length = (central angle / 360°) × 2πr. Sector area = (central angle / 360°) × πr². Inscribed angle = ½ × central angle subtending the same arc. A chord is a line segment with both endpoints on the circle. The diameter is the longest chord. Tangent lines are perpendicular to the radius at the point of tangency.
Coordinate geometry: the distance formula between (x₁,y₁) and (x₂,y₂) is √[(x₂−x₁)² + (y₂−y₁)²]. Midpoint formula: ((x₁+x₂)/2, (y₁+y₂)/2). Slope = rise/run = (y₂−y₁)/(x₂−x₁). Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals (m₁ × m₂ = −1). Equation of a line: y = mx + b (slope-intercept) or y − y₁ = m(x − x₁) (point-slope).
Key Terms:
Quiz Questions:
Q1. A right triangle has legs of length 5 and 12. What is the length of the hypotenuse?
A) 17 B) 13 C) √119 D) 7
Answer: B — This is the 5-12-13 Pythagorean triple: 5² + 12² = 25 + 144 = 169 = 13². Hypotenuse = 13.
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Q2. Two parallel lines are cut by a transversal. Angle A (alternate interior angle on one side) measures 65°. What is the measure of its co-interior (same-side interior) angle on the other parallel line?
A) 65° B) 115° C) 25° D) 90°
Answer: B — Co-interior angles (also called consecutive interior angles or same-side interior angles) are supplementary: they sum to 180°. If one alternate interior angle is 65°, the corresponding co-interior angle on the same transversal is 180° − 65° = 115°.
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Q3. A circle has a radius of 6. What is the area of a sector with a central angle of 120°?
A) 12π B) 4π C) 36π D) 8π
Answer: A — Sector area = (120/360) × π × 6² = (1/3) × 36π = 12π.
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Q4. Line L passes through points (2, 1) and (6, 9). What is the equation of a line perpendicular to L passing through (4, 5)?
A) y = 2x − 3 B) y = −½x + 7 C) y = ½x + 3 D) y = −2x + 13
Answer: B — Slope of L = (9−1)/(6−2) = 8/4 = 2. Perpendicular slope = −1/2. Using point-slope with (4,5): y − 5 = −½(x − 4) → y = −½x + 2 + 5 → y = −½x + 7.
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Q5. In triangle ABC, angle A = 50°, angle B = 70°. What is angle C?
A) 50° B) 70° C) 60° D) 40°
Answer: C — Angles of a triangle sum to 180°: C = 180° − 50° − 70° = 60°.
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