In this section, we will focus on new vocabulary that describe how two angles can be related, also known as angle pair relationships.
ANGLE PAIR #1: COMPLEMENTARY ANGLES
If two angles are complementary angles, then their measures add to \(90^{\circ}\).
If two angles are complementary angles, then their measures add to \(90^{\circ}\).
Let’s look at some examples.
Example 1:
Given \(\angle{JER}\) and \(\angle{REM}\) in the diagram below are complementary angles, find \(m\angle{JER}\).
Example 1:
Given \(\angle{JER}\) and \(\angle{REM}\) in the diagram below are complementary angles, find \(m\angle{JER}\).
Solution:
Since \(\angle{JER}\) and \(\angle{REM}\) are complementary angles, their measures must add up to be \(90^{\circ}\). Therefore:
\(\begin{align*}
m\angle{JER} + m\angle{REM} &= 90^{\circ}\\
m\angle{JER} + 22^{\circ} &= 90^{\circ}\\
m\angle{JER} &= 68^{\circ}
\end{align*}\)
Two angles that are complementary do NOT need to be adjacent. Here’s another example:
Example 2: Given two complementary angles where the measure of the larger angle is three more than twice the smaller angle, find the measure of the larger angle.
Solution:
Let's watch the video for the solution.
Since \(\angle{JER}\) and \(\angle{REM}\) are complementary angles, their measures must add up to be \(90^{\circ}\). Therefore:
\(\begin{align*}
m\angle{JER} + m\angle{REM} &= 90^{\circ}\\
m\angle{JER} + 22^{\circ} &= 90^{\circ}\\
m\angle{JER} &= 68^{\circ}
\end{align*}\)
Two angles that are complementary do NOT need to be adjacent. Here’s another example:
Example 2: Given two complementary angles where the measure of the larger angle is three more than twice the smaller angle, find the measure of the larger angle.
Solution:
Let's watch the video for the solution.
ANGLE PAIR #2: SUPPLEMENTARY ANGLES
If two angles are supplementary, then their measures add up to \(180^{\circ}\).
If two angles are supplementary, then their measures add up to \(180^{\circ}\).
Let’s try an example below.
Example 3:
Given that \(\angle{PAK}\) and \(\angle{ZAK}\) are supplementary. If \(m\angle{PAK}=(4x + 15)^{\circ}\) and \(m\angle{ZAK}=(3x - 10)^{\circ}\), find the measure of the larger angle.
Solution:
Since \(\angle{PAK}\) and \(\angle{ZAK}\) are supplementary, their measures add up to \(180^{\circ}\). So we can write and solve the following equation:
\(\begin{align*}
m\angle{PAK} + m\angle{ZAK} &= 180^{\circ}\\
4x + 15 + 3x - 10 &= 180 & \text{Substitute the measures for both angles}\\
7x + 5 &= 180 & \text{Combine like terms}\\
x &= 25 & \text{Divide both sides of the equation by 2}
\end{align*}\)
Since this problem asks for the measure of the larger angle, we must substitute the value for \(x\) back into both expressions to see which is larger:
\(\begin{align*}
m\angle{PAK} &= (4x + 15)^{\circ} & m\angle{ZAK} &= (3x - 10)^{\circ}\\
&= 4(25) + 15 & &= 3(25) - 10\\
&= 100 +15 & &= 75 - 10\\
m\angle{PAK} &= 115^{\circ} & m\angle{ZAK} &= 65^{\circ}
\end{align*}\)
Since \(115^{\circ}\) is larger than \(65^{\circ}\), the measure of the larger angle is \(115^{\circ}\).
Example 4:
The supplement of an angle is \(27^{\circ}\) more than four times its complement. Find the supplement.
Solution:
Let angle = \(x\)
complement = \(90 - x\)
supplement = \(180 - x\)
\(\begin{align*}
180 - x &= 27 + 4(90 - x)\\
180 - x &= 27 + 360 - 4x\\
180 - x + 4x &= 387 - 4x + 4x\\
180 + 3x &= 387\\
180 - 180 + 3x &= 387 - 180\\
3x &= 207\\
\frac{3x}{3} &= \frac{207}{3}\\
x &= 69
\end{align*}\)
Supplement = \(180 - 69 = 111^{\circ}\)
Example 3:
Given that \(\angle{PAK}\) and \(\angle{ZAK}\) are supplementary. If \(m\angle{PAK}=(4x + 15)^{\circ}\) and \(m\angle{ZAK}=(3x - 10)^{\circ}\), find the measure of the larger angle.
Solution:
Since \(\angle{PAK}\) and \(\angle{ZAK}\) are supplementary, their measures add up to \(180^{\circ}\). So we can write and solve the following equation:
\(\begin{align*}
m\angle{PAK} + m\angle{ZAK} &= 180^{\circ}\\
4x + 15 + 3x - 10 &= 180 & \text{Substitute the measures for both angles}\\
7x + 5 &= 180 & \text{Combine like terms}\\
x &= 25 & \text{Divide both sides of the equation by 2}
\end{align*}\)
Since this problem asks for the measure of the larger angle, we must substitute the value for \(x\) back into both expressions to see which is larger:
\(\begin{align*}
m\angle{PAK} &= (4x + 15)^{\circ} & m\angle{ZAK} &= (3x - 10)^{\circ}\\
&= 4(25) + 15 & &= 3(25) - 10\\
&= 100 +15 & &= 75 - 10\\
m\angle{PAK} &= 115^{\circ} & m\angle{ZAK} &= 65^{\circ}
\end{align*}\)
Since \(115^{\circ}\) is larger than \(65^{\circ}\), the measure of the larger angle is \(115^{\circ}\).
Example 4:
The supplement of an angle is \(27^{\circ}\) more than four times its complement. Find the supplement.
Solution:
Let angle = \(x\)
complement = \(90 - x\)
supplement = \(180 - x\)
\(\begin{align*}
180 - x &= 27 + 4(90 - x)\\
180 - x &= 27 + 360 - 4x\\
180 - x + 4x &= 387 - 4x + 4x\\
180 + 3x &= 387\\
180 - 180 + 3x &= 387 - 180\\
3x &= 207\\
\frac{3x}{3} &= \frac{207}{3}\\
x &= 69
\end{align*}\)
Supplement = \(180 - 69 = 111^{\circ}\)
ANGLE PAIR #3: LINEAR PAIR
If two angles are a linear pair, then they are adjacent supplementary angles that form a straight line. See the diagram below:
If two angles are a linear pair, then they are adjacent supplementary angles that form a straight line. See the diagram below:
Example 5:
If two angles form a linear pair where one is twice the measure of the other, find the measure of each angle.
Solution:
Suppose the measure of one angle is \(x^{\circ}\). Since the measure of the other angle is twice that of one angle, the measure of the other angle would be \((2x)^{\circ}\). Since we are looking at two angles that are a linear pair, we know that their measures must add up to \(180^{\circ}\). So we can set up the following equation:
\(\begin{align*}
x + 2x &= 180^{\circ}\\
3x &= 180^{\circ}\\
x &= 60
\end{align*}\)
Since \(x=60^{\circ}\), then \(2x = 120^{\circ}\). Thus the measure of each angle is: \(60^{\circ}\) and \(120^{\circ}\).
If two angles form a linear pair where one is twice the measure of the other, find the measure of each angle.
Solution:
Suppose the measure of one angle is \(x^{\circ}\). Since the measure of the other angle is twice that of one angle, the measure of the other angle would be \((2x)^{\circ}\). Since we are looking at two angles that are a linear pair, we know that their measures must add up to \(180^{\circ}\). So we can set up the following equation:
\(\begin{align*}
x + 2x &= 180^{\circ}\\
3x &= 180^{\circ}\\
x &= 60
\end{align*}\)
Since \(x=60^{\circ}\), then \(2x = 120^{\circ}\). Thus the measure of each angle is: \(60^{\circ}\) and \(120^{\circ}\).
Sidenote: \(\angle 1\) and \(\angle 4\) are a linear pair, and thus they are supplementary. There are several other pairs of supplementary angles in this diagram - can you name them?
Since vertical angles are created by two straight lines, they have a special relationship:
Since vertical angles are created by two straight lines, they have a special relationship:
If two angles are vertical angles, then they are congruent.
Example 6: Given \(m\angle 4 = (5x - 90)^{\circ}\), \(m\angle 2 = (2x + 30)^{\circ}\), and \(m\angle 3 = (2y)^{\circ}\), solve for \(x\) and \(y\).
Solution:
Let's watch the video for the solution.
Solution:
Let's watch the video for the solution.
Quick Check
1) Given \(\angle A\) and \(\angle B\) are complementary angles. If \(m\angle B = 34^{\circ}\), find \(m\angle A\).
2) Name a linear pair from the diagram below.
1) Given \(\angle A\) and \(\angle B\) are complementary angles. If \(m\angle B = 34^{\circ}\), find \(m\angle A\).
2) Name a linear pair from the diagram below.
3) If \(m\angle 8 = \left (\frac{1}{2}x + 25\right)^{\circ}\), \(m\angle 5 = (11x + 17)^{\circ}\), and \(m\angle 7 = (5y - 1)^{\circ}\), solve for \(x\) and \(y\):
4) Solve for \(x\) in the following:
5) The supplement of an angle is eighteen less than three times its complement. Find the complement.
Quick Check Solutions
Quick Check Solutions