Lesson

	Blueprints for this Lesson:
	Identify and analyze angle pairs. Use knowledge of angle pairs to find the measures of missing angles.

Foundational Knowledge:

Use the information given in each figure to find the measure of the missing angle, labeled x.

Angle D A B is a right angle. Ray A C cuts the angle into two parts. Angle D A C is labeled x degrees and angle C A B is labeled 50 degrees. Obtuse angle Q R P is adjacent to acute angle P R S. These angles are supplementary. The measure of angle Q R P is 120 degrees. Angle Q R S is a straight angle. X represents the measure of angle P R S.

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Framework for Understanding:

You have learned many things about angles so far in this module. Now, let’s focus on the relationships between two angles, sometimes called angle pairs.

adjacent angles A X B and B X C. They share side X B Two angles that share a side are called adjacent angles.

are adjacent angles because they share

and have the same vertex, X.

Important icon It is important to know that while share side they are not considered adjacent. This is because angle A X B is part of angle A X C .

angle J K H and angle H K L shares side K H and the two angles form a straight angle. Two angles that together form a straight angle are called a linear pair.

angle J K H and angle H K L are a linear pair.

angle J K H and angle H K L shares side K H and the two angles are measured 105 degrees and 75 degrees Two angles that add up to 180 degrees are called supplementary angles.

angle J K L is a straight angle that measures 180 degrees. angle J K H and angle H K L are called supplementary angles because their measures add up to 180 degrees.

The measure of angle J K H plus the measure of angle H K L equals the measure of angle J K L.
Next Line. 105 degrees plus 75 degrees equals 180 degrees.

angle J K H is the supplement of angle H K L , and is the supplement of .

If supplementary angles are adjacent, then they are also called a linear pair. Supplementary angles do not have to be adjacent in order to be supplementary. As long as two angles add to be 180 degrees, they are supplementary. Here is another visual example.

Two seperate angels. The measure of angle D E F is 165 degrees. The measure of angle A B C is 15 degrees. what is the measure of angle D E F plus the measure of angle A B C.
Next Line. 165 degrees plus 15 degrees equals 180 degrees.

Because they add up to 180 degrees, angle D E F and angle A B C are supplementary.

angle O P M and angle O P N are complementary and share side P O Two angles that add up to 90 degrees are called complementary angles.

angle M P N is a right angle and measures 90 degrees. The two smaller angles that make up are angle O P N and angle O P M . They are called complementary angles because their measures add up to 90 degrees.

The measure of angle O P N and the measure of angle O P M equals the measure of angle M P N. Next Line. 66 degrees plus 24 degrees equals 90 degrees.

angle O P N is the complement of angle O P M and is the complement of .

Complementary angles do not have to be next to one another to be complementary. As long as two angles add to be 90 degrees, they are complementary. Here is another visual example.

Two separate angles. The measure of angle A B C is 78 degrees. The measure of angle D E F is 12 degrees. What is the measure of angle A B C plus the measure of angle D E F? Next Line. 78 degrees plus 12 degrees equals 90 degrees.

Because they add up to 90 degrees, angle D E F and angle A B C are complementary.

two lines intersect and formed four angles. Counting down clockwise they are angle 1, 2, 3, 4. Two non-adjacent angles that are opposite (across from) each other when two lines intersect are called vertical angles. Pairs of vertical angles are congruent.

In the figure to the right, angle 1 and angle 3 are a pair of vertical angles. Angle 2 and angle 4 are also a pair of vertical angles.

Angle 1 is congruent to angle 3. Angle 2 is congruent to angle 4.

You can use the angle pair relationships above to find measures of missing angles.

Example:

	Use the given information to find the measures of all the angles in the figure. Given:
	Step 1: Make a sketch of this figure on your own paper and label the measure of angle 5.
	Step 2: Use the angle pair relationships to find the measures of the other missing angles. There are many ways to start this problem. You might see a different starting point, but eventually all the angle measures can be found. One starting point is to look for vertical angles in the illustration. Angle 5 and Angle 3 are vertical angles.
	Angle 4 and angle 5 form a linear pair. That tells us they are supplementary angles and their measures must total 180 degrees. We can use subtraction to figure out the measure of angle 4. 180 – 47 = 133
	The measure of angle 1 is given in the drawing. It is a right angle.
	There are several ways to find the measure of angle 2. Angles 1, 2 and 3 make a straight angle totaling 180 degrees. We know that angle 1 is a right angle measuring 90 degrees, so angles 2 and 3 combined must be another 90 degrees. Earlier we found the measure of angle 3 to be 47 degrees. So, if we subtract that from 90 degrees, we have 43.
	Step 3: Answer the question.

Example:

Two lines intersect to form 4 angles. The angles are numbered 1, 2, 3, and 4. Angle 1 is 3 times x plus 8 degrees. Angle 3 is 5 times x minus 20 degrees. Angle 4 is 5 times x plus 4 times y degrees.

Use the given information to find the values of x and y. Find the measures of all four angles in this figure.

Before you get started, study the figure to identify any angle pair relationships.

What do you know?

angle 1 and angle 3 are vertical angles.
angle 2 and angle 4 are vertical angles.
angle 1 and angle 2 are supplementary angles.
angle 1 and angle 4 are supplementary angles.
angle 2 and angle 3 are supplementary angles.
angle 3 and angle 4 are supplementary angles.

Does any of this information help you set up an equation to solve for x?

angle 1 and angle 3 are vertical angles, so the measures of their angles are congruent.

We can write the following equation and solve for x.

3x + 8 = 5x – 20
-2x + 8 = - 20
- 2x = - 28
x = 14

Now, we can find the measures of angles 1 and 3.

The measure of angle 1 equals 3 times x plus 8. Since x equals 14, the measure of angle 1 equals 3 times 14 plus 8, which equals 42 plus 8, which equals 50 degrees. The measure of angle 3 equals 5 times x minus 20. Since x equals 14, the measure of angle 3 equals 5 times 14 minus 20, which equals 70 minus 20, which equals 50 degrees.

Now that we know the value of x, and the measures of angles 1 and 3, we can find the measure of angle 4.

angle 1 and angle 4 are supplementary angles. The sum of the angles is 180 degrees. If m angle 1 = 50 then what is the measure of angle 4?

180 – 50 = 130
the measurement of angle four is one hundred thirty degrees

We have found everything except for the value of y.

We know that (5x + 4y) must equal 130. This can be written as an equation.

5x + 4y = 130

Plug in x and solve for y.

5(14) + 4y	=	130
70 + 4y	=	130
4y	=	60
y	=	15

The answer to this problem is a long one.
x = 14, y = 15, the measurement of angle 1 is fifty degrees , the measurement of angle 3 is fifty degrees , the measurement of angle 2 is one hundred thirty degrees and the measurement of angle 4 is one hundred thirty degrees degrees .

Example:

The measure of an angle is 5 less than 4 times its complement. Find the measure of each angle.

If we let the angle be represented by x, then the complement = 90 – x.
Now translate the sentence into an equation.

The measure of an angle
the measure of an angle is x

is
=

5 less than 4 times its complement.
the other side of the sentence is the other side of the equation

4(90-x) - 5

Now it is time to solve.

x	=	4(90-x) - 5
x	=	360 - 4x - 5
x	=	355 - 4x
5x	=	355
x	=	71

This means that the angle is 71° and its complement is 90 - 71 = 19°.

Move on to the practice page to apply what you have learned here.

1.09 Angle Relationships