Geometry I — Lines, Angles, and Triangles

What This Module Covers

Lines and angle relationships, triangle properties (including the triangle inequality and exterior angle theorem), the Pythagorean theorem with its common triples, special right triangles (45-45-90 and 30-60-90), triangle area, and similar triangles.

Why It Matters on the GRE

Geometry shows up on roughly 15–20% of GRE Quant questions, and triangles alone account for a huge chunk of that. The GRE loves to hide a 30-60-90 triangle inside a diagram without labeling it — so you have to recognize the pattern. Miss it and the problem looks impossible; spot it and it collapses in 30 seconds.

Core Concepts

Angle Types and Relationships

Most test-takers remember supplementary angles but blank on the parallel lines rules under pressure. Lock these in before test day.

Must Know: Supplementary angles sum to 180°. Complementary angles sum to 90°. Vertical angles (formed when two lines cross) are always equal to each other.

When a transversal cuts across two parallel lines, four angle relationships kick in simultaneously:

• Corresponding angles (same position at each intersection) are equal.
• Alternate interior angles (on opposite sides of the transversal, between the parallel lines) are equal.
• Alternate exterior angles (on opposite sides, outside the parallel lines) are equal.
• Co-interior (same-side interior) angles are supplementary — they add to 180°.

Must Know: On the GRE, if you see two parallel lines crossed by a transversal, there are really only two distinct angle measures — every angle is either x or (180° − x). Label one angle and you know them all.

Example — Parallel Lines:

1. Two parallel lines are cut by a transversal. One angle measures 65°.
2. The alternate interior angle on the other parallel line is also 65° (alternate interior angles are equal).
3. The co-interior angle on the same side is 180° − 65° = 115°.
4. Every other angle at both intersections is either 65° or 115°. Done.

Triangle Fundamentals

Must Know: The three interior angles of any triangle always sum to 180°. This is non-negotiable — use it to find the missing angle whenever two are given.

The Triangle Inequality says that any side must be strictly less than the sum of the other two sides, and strictly greater than their difference. If two sides are 5 and 9, the third side must satisfy: |9 − 5| < x < 9 + 5, which means 4 < x < 14. The GRE tests both bounds — students almost always remember the upper bound and forget the lower.

The Exterior Angle Theorem is a GRE shortcut that most students overlook. When you extend one side of a triangle, the exterior angle formed equals the sum of the two non-adjacent interior angles.

Must Know: Exterior angle = sum of the two remote interior angles. Skip the detour through 180°.

Example — Exterior Angle:

1. A triangle has interior angles of 50°, 70°, and x°.
2. An exterior angle is formed by extending the side opposite the 50° angle.
3. Exterior angle = 50° + 70° = 120°. (You didn't even need to find x.)

Triangle types to recognize on sight:

• Equilateral: all sides equal, all angles 60°.
• Isosceles: two equal sides, two equal base angles.
• Scalene: all sides and angles different.
• Right: one angle is exactly 90°.

Pythagorean Theorem

Must Know: In a right triangle, a² + b² = c², where c is the hypotenuse (the side opposite the right angle — always the longest side). This theorem only applies to right triangles.

Memorize these Pythagorean triples — the GRE uses them constantly:

When you see a right triangle with legs 6 and 8, don't compute — recognize the 3-4-5 pattern scaled by 2. The hypotenuse is 10.

Special Right Triangles

These two triangles appear constantly and the GRE rarely labels them explicitly. You need to recognize the angle pattern and deploy the ratio from memory.

Must Know — 45-45-90: Sides are in ratio x : x : x√2. The two legs are equal; the hypotenuse is leg × √2. If legs = 5, hypotenuse = 5√2.

Must Know — 30-60-90: Sides are in ratio x : x√3 : 2x. Shortest side (opposite 30°) = x. Middle side (opposite 60°) = x√3. Hypotenuse (opposite 90°) = 2x.

Example — 30-60-90:

1. A right triangle has a hypotenuse of 12 and one angle of 30°.
2. Hypotenuse = 2x → 2x = 12 → x = 6.
3. Short leg (opposite 30°) = x = 6.
4. Long leg (opposite 60°) = x√3 = 6√3.

Example — 45-45-90:

1. An isosceles right triangle has legs of length 7.
2. Hypotenuse = 7√2. No calculation needed — apply the ratio directly.
3. Verification: 7² + 7² = 49 + 49 = 98 = (7√2)². Confirmed.

Triangle Area and Similar Triangles

Must Know: Area = (1/2) × base × height. The height must be perpendicular to the base — it is not necessarily a side of the triangle.

For similar triangles, if the ratio of corresponding sides is k, the ratio of their areas is k². This is a frequent GRE trap: students apply the side ratio to the area directly instead of squaring it.

Must Know: Similar triangles have the same angles. Use AA (Angle-Angle) to prove similarity — if two angles match, the third must also match (since all three sum to 180°).

Example — Similar Triangles:

1. Two similar triangles have corresponding sides in ratio 3:5.
2. The area of the smaller triangle is 18.
3. Area ratio = (3/5)² = 9/25.
4. Area of larger triangle = 18 × (25/9) = 50.

Common Traps

• Pythagorean theorem on non-right triangles: The GRE will show you a triangle without marking the right angle and let you assume it exists. Always confirm a right angle before using a² + b² = c².
• Height ≠ side: The area formula requires a perpendicular height. In an obtuse triangle, the height may fall outside the triangle entirely. Draw it out.
• Hypotenuse confusion: The hypotenuse is always opposite the right angle — not just any long-looking side.
• Missing the special triangle pattern: A problem might give you a 60° angle and a hypotenuse and expect you to apply 30-60-90. If you don't recognize the flag, you'll reach for the Pythagorean theorem and get stuck in ugly algebra.
• Triangle inequality lower bound: If sides are 3 and 11, the third side must be greater than 8 (not just less than 14). Both bounds are testable.
• Similar triangle area trap: If sides scale by k, area scales by k² — not k. Squaring the ratio is mandatory.

GRE Strategy

• When you see a right triangle, immediately check for a Pythagorean triple before doing any algebra. This saves 60–90 seconds per problem.
• If a problem mentions one angle of 30°, 45°, or 60° in a right triangle, write the special triangle ratio immediately — it almost always unlocks the solution path.
• On parallel-lines problems, label one angle and use the "x or 180 − x" rule to fill in every other angle in the diagram before reading the question.
• For triangle area problems, always identify the base-height pair explicitly. Circle the right angle marker if one exists; otherwise draw a dotted perpendicular height.
• When a problem says "what are the possible values of side x," compute both bounds from the triangle inequality and write the full inequality before answering.

Worked Example

Question:

In a 30-60-90 triangle, the hypotenuse has length 10. What is the length of the shorter leg?

(A) 5 (B) 5√2 (C) 5√3 (D) 10/√3 (E) 10√3

Solution:

Step 1 — Identify the ratio. In a 30-60-90 triangle, sides are in ratio x : x√3 : 2x, where 2x is the hypotenuse.

Step 2 — Solve for x. Hypotenuse = 2x = 10 → x = 5.

Step 3 — Identify the shorter leg. The shorter leg is opposite the 30° angle, which equals x = 5.

Answer: (A) 5

Step 4 — Verify. Sides would be 5, 5√3, and 10. Check: 5² + (5√3)² = 25 + 75 = 100 = 10². Correct.

Why the other choices are wrong:

• (B) 5√2 — This is the hypotenuse of a 45-45-90 triangle with legs of 5. You applied the wrong special triangle.
• (C) 5√3 — This is the longer leg (opposite 60°), not the shorter one. You grabbed the right triangle but the wrong side.
• (D) 10/√3 — This comes from dividing the hypotenuse by √3 without a clear geometric basis. It's a distractor that looks algebraically reasonable but doesn't correspond to any side of this triangle.
• (E) 10√3 — This exceeds the hypotenuse, which is impossible for any leg of a right triangle. Any leg must be shorter than the hypotenuse.