Triangles

SAT Math Overview. Topic: 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.

 

Triangle. A three-sided figure with three interior angles and three vertices. The sum of the three interior angles measures is 180°.

A three sided polygon that has three angles, vertices, and sides. The sum of the interior angles of a triangle is equal to 180°.

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. 

Acute Triangle. A triangle whose three interior angles are not greater than 90° in measure.

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

Obtuse Triangle. A triangle, one of whose interior angles is greater than 90° in measure.

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

Right Triangle. A triangle with one right angle and two acute angles. The relationship among the sides is summarized by the Pythagorean Theorem.

  • A triangle two congruent angles is an isosceles triangle. 

Isosceles Triangle. A triangle with two congruent sides and congruent base angles.

  • A triangle three congruent angles is an equilateral triangle. 

Equilateral Triangle. A triangle with three congruent sides and three congruent interior angles.

To learn more about classifying triangles by angle measures, click on this link. It is a presentation that goes over this topic.

 

SAT Skill: Triangle Properties

Example 1

Given the triangle below, find x.

An isosceles triangle.

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.

A system of linear equations.

Use the Substitution Method to solve for x:

The solution to a system of linear equations.

Example 2

Given the triangle below, find z.

Two triangles each of which have a vertex on the center of the circle and two sides on the circle itself.

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:

Solving a system of linear equations.

Now use the value for x and the properties of isosceles triangles to solve for z:

Solving a system of linear equations.

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. 

Scalene Triangle. A triangle with three unequal sides.

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

Isosceles Triangle. A triangle with two congruent sides and congruent base angles.

 

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

Equilateral Triangle. A triangle with three congruent sides and three congruent interior angles.

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

Right Triangle. A triangle with one right angle and two acute angles. The relationship among the sides is summarized by the Pythagorean Theorem.

To learn more about classifying triangles by side lengths, click on this link. It is a presentation that goes over this topic.

SAT Skill: Triangle Properties

Example 1

Given this triangle, what is the length of side AB?

A triangle with one vertex at the center of a circle forming a right angle and the other two vertices on the circle itself, resulting in an isosceles right triangle.

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:

Using the Pythagorean Theorem to find the length of the hypotenuse.

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?

Two triangles that share a side.

Use the Exterior Angle Theorem to generate another equation:

Writing x equals y plus 30 as x minus y equals 30.

Use the two equations to solve a system:

Solving a linear system of equations.

 

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