Introduction: Some angles can be classified according to their positions or measurements in relation to other angles. We examine three types: complementary, supplementary, and vertical angles.

Definitions:

Complementary angles are two angles with a sum of 90º.

Supplementary angles are two angles with a sum of 180º.

Vertical angles are two angles whose sides form two pairs of opposite rays. We can think of these as opposite angles formed by an X.

The Lesson:

In the triangle shown below, the angles A and B are complementary because they have a sum of 90º. This is obvious since angle C is 90º and the other two angles must have a sum of 90º so that the three angles in the triangle together have a sum of 180º. It is always true that the two acute angles in a right triangle are complementary.

In the diagram below, angles 1 and 2 are supplementary because they form the straight line QP. This is equivalent to 180º.

In the diagram below, angles 1 and 2 are vertical because they form an X and are opposite angles in that X.

Let's Practice:

1. Using the diagram below, list several pairs of supplementary and vertical angles.

Angle 1 is vertical with .

Angle 2 is vertical with . In each case these pairs of angles form an X.

and are supplementary because they form the straight line FC.

and are supplementary because they form the straight line AD.

and are vertical.

and are supplementary because they form the straight line FC.

2. In the diagram below, both angles 1 and 2 are supplementary to angle 3.

This means that:

Subtracting these equations gives us

Theorem: Vertical angles are equal (have the same measure).

3. In a right triangle PQR (not shown) with angle Q a right angle, suppose we have . What is the value of x and what are the measures of angles P and R?

Since angles P and R are complementary in a right triangle, we must have

Substituting in our given information about angles P and R

Consequently

4. In the diagram below suppose that . What is the value of x and what is the measure of angle 3?

Since angles 1 and 2 are vertical, they must have the same measure.

Therefore we have

Solving this equation,

we get x = 5

Substituting,

angle 1 has measure 4(5) = 20º and the same is true of

angle 2: 2(5) + 10 = 20º

We use this information to find angle 3 which is supplementary to both angles 1 and 2.

angle 3 has measure 160º because supplementary angles have a sum of 180º