# Triangles

## Overview

Expect to see questions that test your understanding of triangles and their properties. In this section we’ll cover the following:

• Triangle Basics
• Classifying Triangles by Angle
• Classifying Triangles by Side
• Triangle Theorems

Let’s look at each of these in more detail:

### Triangle Basics

What is a triangle? Look at these definitions.

Despite their common properties, triangles come in different sizes and angle measures. and they can be categorized by side lengths and angle measures. To review triangle basics, click on this link. It is a presentation that goes over this topic.

### Classifying Triangles by Angles

Depending on the angle measures that make up a triangle, the result is a specific type of triangle. Here are the different classifications of triangles by angle:

• A triangle with three acute angles is called an acute triangle

• A triangle with one acute angle is called an obtuse triangle. A triangle can only have one obtuse angle.

• A triangle with a right angle is called a right triangle. A triangle can only have one right angle.

• A triangle two congruent angles is an isosceles triangle

• A triangle three congruent angles is an equilateral triangle

 SAT Skill: Triangle Properties Example 1 Given the triangle below, find x. In this triangle, sides AC and BC are congruent, so this is an isosceles triangle. This means that the Angle CAB is also x. As a result we get this system of equations. Use the Substitution Method to solve for x: Example 2 Given the triangle below, find z. Notice that each triangle has two sides that are radii of the circle. That means that each of the triangles is isosceles. Use the system of equations to solve for y: Now use the value for x and the properties of isosceles triangles to solve for z: Note: This means that triangle ABC is an equilateral triangle.

### Classifying Triangles by Side Length

In addition to classifying triangles by the type of angles that make it up, it’s also possible to classify triangles by their side lengths.

• A triangle with three sides that are of different lengths is called a scalene triangle

• A triangle with two sides that are the same length is called an isosceles triangle

• A triangle with three sides that are the same length is called an equilateral triangle

• A triangle whose sides align with the Pythagorean Theorem is a right triangle

 SAT Skill: Triangle Properties Example 1 Given this triangle, what is the length of side AB? Notice that this triangle is a right triangle, as indicated by angle ACB. Also, the two legs of this right triangle are made up of radii of the circle. This means that side AC is also 8. Use the Pythagorean Theorem to find the length of side AB:

### Triangle Theorems

Make sure you are familiar with these key triangle theorems. They will often be part of solving a particular SAT problem involving triangles.

Click on this link to see a slide show of these theorems.

 SAT Skill: Triangle Properties Example 1 Given these triangles, what is the value of x? Use the Exterior Angle Theorem to generate another equation: Use the two equations to solve a system: