Angles

SAT Math Overview. Topic: Angles

Overview

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

  • Angle Basics
  • Classifying Angles by Size
  • Classifying Angles Relationships
  • Identifying Angle Properties in Other Geometric Figures

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

Angle Basics

What is an angle? Look at this definition.

Angle. The figure formed from two rays, lines, or segments extending from the same point, known as the vertex.

To review angle basics, click on this link. It is a presentation that goes over this topic.

Classifying Angles by Size

Angles range from 0° to 360°. Depending on the measure of the angle, it can fall into several categories of angles. 

  • An angle whose measure is less than 90° is an acute angle. 

Acute Angle. An angle whose measure is less than 90°.

  • An angle greater than 90° and less than 180° is an obtuse angle.

Obtuse Angle. An angle whose measure is greater than 90°.

  • An angle that measures 90° is a right angle.

Right Angle. An angle whose measure is 90°.

  • An angle greater than 180° and less than 360° is called a reflex angle. 

Reflex Angle. An angle whose measure is greater than 180° and less than 360°.

  • An angle that is 180° is called a straight angle. 

Straight Angle. An angle whose measure is 180°.

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

Classifying Angles Relationships

Angles can be classified by their angle measure. Often angles are associated with other angles. For example, two angles can share a side:

Two adjacent angles, A and B, that share a side.

Angles make up the interior of polygons:

Triangle ABC.

 

Or, when two lines intersect, four angles are formed:

Two intersecting lines forming angles A, B, C, and D.

Take a look at this link to see a presentation on these different angle relationships. In particular, make sure you understand the following types of angle relationships:

  • Complementary Angles
  • Supplementary Angles
  • Vertical Angles

A special case of angle relationships happens when two parallel lines are cut by a transversal.

Parallel Lines Cut by a Transversal. Two lines, L and M, cut by transversal N.

Click on this link to see a presentation that shows all the angles formed by this configuration. Make a note of the congruent and supplementary angles formed.

 

SAT Skill: Angle Properties

Example 1

Given the triangle below, find x.

A right triangle formed by two perpendicular lines.

Because the lines are perpendicular to each other, then the triangle is a right triangle, as shown here.

A right triangle formed by two perpendicular lines.

That means that the two acute angles of the right triangle are complementary. So, we can write an equation to solve for x:

Solving the equation 38 plus x equals 90.

Example 2

Given the parallel lines below, find x.

Parallel lines cut by a transversal. Two supplementary angles are labeled.

The two parallel lines are cut by a transversal. The two angles shown are supplementary to each other.

Parallel lines cut by a transversal. Two supplementary angles are labeled.

Now you can write an equation and solve for x:

Solving the equation x plus 121 equals 180.

Identifying Angle Properties in Other Geometric Figures

Make sure you are familiar with key triangle theorems that relate to angle measures. 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 congruent triangles.

Given that two corresponding sides and an included angle are congruent, then we can conclude that the two angles are congruent using the SAS Theorem.

 

Two congruent triangles.

The exterior angle (115°) is supplementary to angle ABC. This corresponding angle is found on the second triangle.

Two congruent triangles.

We can now write and solve an equation for x:

Solving the equation x plus 65 plus 37 equals 180.

 

Example 2

Given these triangles, what is the value of x?

Two triangles that share a side.

Use the Exterior Angle Theorem to generate another equation:

The solution to the equation x = y + 30.

Use the two equations to solve a system:

The solution to a system of linear equations.

 

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