Triangles — Fundamentals
1. Core Idea
The triangle is the most-tested figure on the GRE. Four facts generate nearly every question:
- Angles sum to 180 degrees.
- Area = (1/2) x base x height, where the height is perpendicular to that base.
- The largest angle faces the largest side.
- The triangle inequality constrains which side lengths are possible.
2. Must-Know Rules
| Concept | Rule |
|---|---|
| Angle sum | 180 degrees |
| Exterior angle | equals the sum of the two remote interior angles |
| Area | (1/2) x base x height |
| Area (Heron) | sqrt[s(s-a)(s-b)(s-c)], s = (a+b+c)/2 |
| Triangle inequality | |b - c| < a < b + c |
| Side-angle correspondence | largest side is opposite the largest angle |
| Perimeter | a + b + c |
| Median | joins a vertex to the midpoint of the opposite side |
The triangle inequality in usable form. For sides of length 5 and 9, the third side x satisfies:
9 - 5 < x < 9 + 5, i.e. 4 < x < 14.
Note the inequalities are strict — x = 4 or x = 14 gives a degenerate (flat) triangle.
Triangle classification:
| By sides | By angles | ||
|---|---|---|---|
| Scalene | all sides different | Acute | all angles < 90 |
| Isosceles | two sides equal | Right | one angle = 90 |
| Equilateral | all three equal | Obtuse | one angle > 90 |
Exterior angle theorem. If the interior angles are A, B and C, the exterior angle at C equals A + B. This saves a step constantly.
A median splits a triangle into two triangles of equal area (same base length, same height).
3. Worked Examples
Example 1 — Triangle inequality range
Two sides of a triangle are 7 and 11. How many integer values are possible for the third side?
11 - 7 < x < 11 + 7, so 4 < x < 18.
Integers from 5 to 17 inclusive: 17 - 5 + 1 = 13 values.
Example 2 — Exterior angle
In triangle ABC, angle A = 50 and angle B = 65. What is the exterior angle at C?
Exterior angle at C = A + B = 50 + 65 = 115 degrees.
(Check: interior C = 180 - 115 = 65, and 50 + 65 + 65 = 180. Correct.)
Example 3 — Area with a non-obvious height
A triangle has sides 13, 14 and 15. Find its area.
Use Heron's formula. s = (13 + 14 + 15)/2 = 21.
Area = sqrt[21 x (21-13) x (21-14) x (21-15)] = sqrt[21 x 8 x 7 x 6]
= sqrt[7056] = 84.
4. GRE Traps
- Using a slanted side as the height. The height must be perpendicular to the chosen base.
- Forgetting the triangle inequality has a lower bound too. Students remember a + b > c but forget |a - b| < c.
- Assuming a triangle is isosceles or right because it looks that way.
- Confusing the exterior angle with the interior one.
- Assuming the largest side is opposite the smallest angle. It is the opposite.
- Height falling outside the triangle. In an obtuse triangle the foot of the altitude can lie beyond the base — the formula still holds.
- Adding angles to 360. A triangle's interior angles sum to 180.
5. Speed Tricks
- Third side range: difference < x < sum. One line, always.
- Exterior angle = sum of the two remote interiors. Faster than computing the interior angle first.
- Area ratios: two triangles with the same height have areas in the ratio of their bases; with the same base, in the ratio of their heights.
- Heron only when nothing else works — it involves a square root and is slow. Look for a right angle first.
- In any triangle, rank the sides to rank the angles and vice versa. Many quantitative comparison questions need nothing more.
6. Self-Check
Q1. Two sides of a triangle are 6 and 10. What is the range of the third side?
Q2. A triangle has angles in the ratio 2 : 3 : 4. Find the largest angle.
Q3. A triangle has base 12 and area 42. What is its height to that base?
Answers
A1. 10 - 6 < x < 10 + 6, so 4 < x < 16.
A2. 9 parts = 180, one part = 20. Largest = 4 x 20 = 80 degrees.
A3. 42 = (1/2)(12)h -> 6h = 42 -> h = 7.
7. One-Line Summary for the Board
Angles sum to 180. Third side is between the difference and the sum. Biggest side faces the biggest angle.