Geometry (A): READ: Angle Pairs Reading

Angle Pairs

Learning Objectives

Understand and identify complementary angles.
Understand and identify supplementary angles.
Understand and utilize the Linear Pair Postulate.
Understand and identify vertical angles.

Introduction

In this lesson you will learn about special angle pairs and prove the vertical angles theorem, one of the most useful theorems in geometry.

Complementary Angles

A pair of angles are Complementary angles if the sum of their measures is $90^\circ$ .

Complementary angles do not have to be congruent to each other. Rather, their only defining quality is that the sum of their measures is equal to the measure of a right angle: $90^\circ$ . If the outer rays of two adjacent angles form a right angle, then the angles are complementary.

Example 1

The two angles below are complementary. $m\angle{GHI}=x$ . What is the value of $x$ ?

Since you know that the two angles must sum to $90^\circ$ , you can create an equation. Then solve for the variable. In this case, the variable is $x$ .

$34+x &= 90 \\ 34 + x -4 & = 90-34\\ x & =56$

: Thus, the value of $x$ is $56^\circ$ .

Example 2

The two angles below are complementary. What is the measure of each angle?

This problem is a bit more complicated than the first example. However, the concepts are the same. If you add the two angles together, the sum will be $90^\circ$ . So, you can set up an algebraic equation with the values presented.

$(7r+6) + (8r+9) = 90$

The best way to solve this problem is to solve the equation above for $r$ . Then, you must substitute the value for $r$ back into the original expressions to find the value of each angle.

$(7r+6)+(8r+9) &= 90\\ 15r + 15 &= 90 \\ 15r + 15 - 15 &= 90-15 \\ 15r &= 75 \\ \frac{15r}{15} &= \frac{75}{15} \\ r &= 5$

The value of $r$ is $5$ . Now substitute this value back into the expressions to find the measures of the two angles in the diagram.

$7r& + 6 & 8r& + 9\\ 7(5)& + 6 & 8(5)& + 9\\ 35& + 6 & 40& + 9\\ 41& & 49&$

$m\angle{JKL}=41^{\circ}$ and $m\angle{GHI}=49^{\circ}$ . You can check to make sure these numbers are accurate by verifying if they are complementary.

$41 + 49 = 90$

Since these two angle measures sum to $90^\circ$ , they are complementary.

Supplementary Angles

Two angles are supplementary if their measures sum to $180^\circ$ .

Just like complementary angles, supplementary angles need not be congruent, or even touching. Their defining quality is that when their measures are added together, the sum is $180^\circ$ . You can use this information just as you did with complementary angles to solve different types of problems.

Example 3

The two angles below are supplementary. If $m\angle{MNO}=78^{\circ}$ , what is $m\angle{PQR}$ ?

This process is very straightforward. Since you know that the two angles must sum to $180^\circ$ , you can create an equation. Use a variable for the unknown angle measure and then solve for the variable. In this case, let's substitute $y$ for $m\angle{PQR}$ .

$78+y&=180\\ 78+y-78&=180-78\\ y &=102$

So, the measure of $y=102$ and thus $m\angle{PQR}=102^{\circ}$ .

Example 4

What is the measure of two congruent, supplementary angles?

There is no diagram to help you visualize this scenario, so you’ll have to imagine the angles (or even better, draw it yourself by translating the words into a picture!). Two supplementary angles must sum to $180^\circ$ . Congruent angles must have the same measure. So, you need to find two congruent angles that are supplementary. You can divide $180^\circ$ by two to find the value of each angle.

$180 \div 2 = 90$

Each congruent, supplementary angle will measure $90^\circ$ . In other words, they will be right angles.

Linear Pairs

Before we talk about a special pair of angles called linear pairs, we need to define adjacent angles. Two angles are adjacent if they share the same vertex and one side, but they do not overlap. In the diagram below, $\angle{PQR}$ and $\angle{RQS}$ are adjacent.

However, $\angle{PQR}$ and $\angle{PQS}$ are not adjacent since they overlap (i.e. they share common points in the interior of the angle).

Now we are ready to talk about linear pairs. A linear pair is two angles that are adjacent and whose non-common sides form a straight line. In the diagram below, $\angle{MNP}$ and $\angle{PNO}$ are a linear pair. Note that $\overleftrightarrow{MO}$ is a line.

Linear pairs are so important in geometry that they have their own postulate.

Linear Pair Postulate: If two angles are a linear pair, then they are supplementary.

Example 5

The two angles below form a linear pair. What is the value of each angle?

If you add the two angles, the sum will be $180^\circ$ . So, you can set up an algebraic equation with the values presented.

$(3q)+(15q+18)=180$

The best way to solve this problem is to solve the equation above for $q$ . Then, you must plug the value for $q$ back into the original expressions to find the value of each angle.

$(3q) + (15q + 18) & = 180\\ 18q+18 &=180\\ 18q &= 180-18\\ 18q &=162\\ \frac{18q}{18} &=\frac{162}{18}\\ q &=9$

The value of $q$ is $9$ . Now substitute this value back into the expressions to determine the measures of the two angles in the diagram.

$3q& & 15q& + 18\\ 3(9)& & 15(9)& + 18\\ 27& & 135& + 18\\ & & 153&$

The two angles in the diagram measure $27^\circ$ and $153^\circ$ . You can check to make sure these numbers are accurate by verifying if they are supplementary.

$27+153=180$

Vertical Angles

Now that you understand supplementary and complementary angles, you can examine more complicated situations. Special angle relationships are formed when two lines intersect, and you can use your knowledge of linear pairs of angles to explore each angle further.

Vertical angles are defined as two non-adjacent angles formed by intersecting lines. In the diagram below, $\angle{1}$ and $\angle{3}$ are vertical angles. Also, $\angle{4}$ and $\angle{2}$ are vertical angles.

Suppose that you know $m\angle{1}=100^{\circ}$ . You can use that information to find the measurement of all the other angles. For example, $\angle{1}$ and $\angle{2}$ must be supplementary since they are a linear pair. So, to find $m\angle{2}$ , subtract $100^\circ$ from $180^\circ$ .

$m\angle{1}+m\angle{2} &=180\\ 100+ m\angle{2} &= 180\\ m\angle{2}& = 180-100\\ m\angle{2}& = 80$

So $\angle{2}$ measures $80^\circ$ . Knowing that angles $2$ and $3$ are also supplementary means that $m\angle{3}=100^{\circ}$ , since the sum of $100^\circ$ and $80^\circ$ is $180^\circ$ . If angle $3$ measures $100^\circ$ , then the measure of angle $4$ must be $80^\circ$ , since $3$ and $4$ are also supplementary. Notice that angles $1$ and $3$ are congruent $(100^\circ)$ and $2$ and $4$ are congruent $(80^\circ)$ .

The Vertical Angles Theorem states that if two angles are vertical angles then they are congruent.

We can prove the vertical angles theorem using a process just like the one we used above. There was nothing special about the given measure of $\angle{1}$ . Here is proof that vertical angles will always be congruent: Since $\angle{1}$ and $\angle{2}$ form a linear pair, we know that they are supplementary: $m\angle{1}+m\angle{2}=180^{\circ}$ . For the same reason, $\angle{2}$ and $\angle{3}$ are supplementary: $m\angle{2}+m\angle{3}=180^{\circ}$ . Using a substitution, we can write $m\angle{1}+m\angle{2}=m\angle{2}+m\angle{3}$ . Finally, subtracting $m\angle{2}$ on both sides yields $m\angle{1}=m\angle{3}$ . Or, by the definition of congruent angles, $\angle{1} \cong \angle{3}$ .

Use your knowledge of vertical angles to solve the following problem.

Example 6

What is $m\angle{STU}$ in the diagram below?

Using your knowledge of intersecting lines, you can identify that $\angle{STU}$ is vertical to the angle marked $18^\circ$ . Since vertical angles are congruent, they will have the same measure. So, $m \angle{STU}$ is also equal to $18^\circ$ .

Lesson Summary

In this lesson, we explored angle pairs. Specifically, we have learned:

How to understand and identify complementary angles.
How to understand and identify supplementary angles.
How to understand and utilize the Linear Pair Postulate.
How to understand and identify vertical angles.

The relationships between different angles are used in almost every type of geometric application. Make sure that these concepts are retained as you progress in your studies.

The following questions are for your own benefit. The answers are listed below in order for you to check your work.

Review Questions

Find the measure of the angle complementary to if
1. $45^\circ$
2. $82^\circ$
3. $19^\circ$
4. $z^\circ$
Find the measure of the angle supplementary to if
1. $45^\circ$
2. $118^\circ$
3. $32^\circ$
4. $x^\circ$
Find $m\angle{ABD}$ and $m\angle{DBC}$ .
Given $m\angle{EFG} = 20^\circ$ , Find $m\angle{HFG}$ .

Use the diagram below for exercises 5 and 6. Note that $\overline{NK} \perp$ $\overleftrightarrow{IL}$ .
Identify each of the following (there may be more than one correct answer for some of these questions).
1. Name one pair of vertical angles.
2. Name one linear pair of angles.
3. Name two complementary angles.
4. Nam two supplementary angles.
Given that , find
1. $m\angle{JNK}$ .
2. $m\angle{KNL}$ .
3. $m\angle{MNL}$ .
4. $m\angle{MNI}$ .

Review Answers

1. $45^\circ$
2. $8^\circ$
3. $81^\circ$
4. $(90 - z)^\circ$
1. $135^\circ$
2. $62^\circ$
3. $148^\circ$
4. $(180 - x)^\circ$
$m \angle{ABD} = 73^\circ$ , $m \angle{DBC} = 107^\circ$
$m \angle{HFG} = 70^\circ$
1. $\angle{JNI}$ and $\angle{MNL}$ (or $\angle{INM}$ and $\angle{JNL}$ also works);
2. $\angle{INM}$ and $\angle{MNL}$ (or $\angle{INK}$ and $\angle{KNL}$ also works);
3. $\angle{INK}$ and $\angle{JNK}$ ;
4. same as (b) $\angle{INM}$ and $\angle{MNL}$ (or $\angle{INK}$ and $\angle{KNL}$ also works).
1. $27^\circ$
2. $90^\circ$
3. $63^\circ$
4. $117^\circ$

Last modified: Monday, June 28, 2010, 12:07 PM