Table of Contents

1. Introduction
2. Common Right Triangles
   • 45-45-90 Triangles
   • 30-60-90 Triangles
3. Similar Triangles

Introduction

The triangle is a polygon that frequently shows up on the GMAT. While all triangle problems can be solved with the information provided, it is not realistic to derive the geometric proofs necessary to solve some problems given the time constraints on the test. The following rules should be memorized because they can save you precious time during the actual exam. Proofs exist for all of these rules and can be found on the internet.

1. The sum of the interior angles of a triangle is 180 degrees
2. An angle lying opposite the greatest side is also the greatest angle; the inverse is also true
3. Angles lying opposite of equal sides are also equal; the inverse is also true.
4. An exterior angle of a triangle is equal to the sum of the non-supplementary interior angles.
5. Any side of a triangle must be less than the sum of the two other sides
6. Any side of a triangle must be more than the difference of the two other sides.
7. Pythagorem theorem (applicable to right triangles only):

Example

Angle bdc (2x) is an exterior angle of triangle abd and has an angle measure equal to the sum of the remote interior angles abd and dab.

Example

What are the possible values of x?

Since the length of x must be less than the sum of the other two sides. That means x < (9 + 3) ⇒ x < 12
Also, the length of x must be greater than the difference of the other two sides. That means x > (9 – 3) ⇒ x > 6

Therefore 6 < x < 12

Common Right Triangles

The following lists some common right triangle side ratios. While all triangles involving the ratios below are right triangles and can be solved using the pythagorem theorem, memorizing these key ratios is much easier and can save a lot of time on the actual exam.

45-45-90 Triangles

This angle combination is very common on the GMAT. Memorize the ratio of the triangle sides to quickly solve any problem involving this type of triangle.

30-60-90-Triangles

Another very common triangle – memorize this ratio to easily bank these problems

Similar Triangles

Two triangles are similar if they are proportional to each other. If two triangles have at least two angles in common, then we know that the two triangles are similar. (We can deduce the third angle is the same because the sum of angles in a triangle must = 180.)

Example

In the figures above, the area of the triangle on the right is twice the area of the triangle on the left. Find A in terms of a.

Solution

First, set the areas of the two triangles equal to each other:

Reduce the equation above:

From the properties of similar angles, we get:

Plug this into the first equation:

Simplify:

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