1.11: Angle Pair Relationships
Learning Objectives
- Define complementary angles, supplementary angles, adjacent angles, linear pairs, and vertical angles.
- Use angle pair relationships to write and solve equations
- Apply the Linear Pair Postulate and the Vertical Angles Theorem.
Complementary Angles
Two angles are complementary angles if the sum of their measures is \begin{align*}90^\circ\end{align*}.
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: \begin{align*}90^\circ\end{align*}. If the outer rays of two adjacent angles form a right angle, then the angles are complementary.
- The measures of complementary angles add up to ___________.
Example 1
The two angles below are complementary. \begin{align*}m\angle GHI=x\end{align*}. What is the value of \begin{align*}x\end{align*}?
Since you know that the two angles must sum to \begin{align*}90^\circ\end{align*}, you can create an equation, then solve for the variable. In this case, the variable is \begin{align*}x\end{align*}:
Since the angles are complementary, \begin{align*}m\angle LKJ + m\angle GHI = 90^\circ\end{align*}
Substitute:
\begin{align*}34^\circ + \ x &= \ 90^\circ\\ – 34^\circ{\;\;\;\;\;\;\;} &{\;\;} -34^\circ\\ x&=\ 56^\circ \end{align*}
The measure of \begin{align*}\angle GHI= 56^\circ\end{align*}.
Example 2
The two angles below are complementary. What is the measure of each angle?
If the angles are complementary, then their measures add up to __________.
The best way to solve this problem is to set up an equation where the two angle expressions sum to \begin{align*}90^\circ\end{align*}. Then solve the equation for \begin{align*}r\end{align*}.
\begin{align*}( \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;} ) + ( \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;} ) = 90^\circ\end{align*}
To find the measure of each angle, you must substitute the value for \begin{align*}r\end{align*} back into the original expressions to find the value of each angle.
\begin{align*}(8r + 9 )+ (7r + 6) & \ = \ 90\\ 15r+15& \ = \ 90\\ \ -15&{\;\;\;}-15\\ 15r & \ = \ 75\\ r & \ = \ 5\end{align*}
The value of \begin{align*}r\end{align*} is 5. Now substitute this value back into both angle expressions to find the measures of the two angles in the diagram:
\begin{align*}&8r+9 && 7r+6\\ &8(5)+ 9 && 7(5)+ 6\\ &40+9 && 35+6\\ &49 && 41\end{align*}
\begin{align*}m\angle GHI = 49^\circ\end{align*} and \begin{align*}m\angle JKL = 41^\circ\end{align*}. You can check to make sure these numbers are accurate by verifying that they are complementary (add up to \begin{align*}90^\circ\end{align*}):
\begin{align*}49 + 41 = 90\end{align*}
Since these two angle measures sum to \begin{align*}90^\circ\end{align*}, they are complementary.
Reading Check:
1. When angles are complementary, what does their sum equal?
\begin{align*}{\;}\end{align*}
\begin{align*}{\;}\end{align*}
2. If you know that the two angles below are complementary, how would you solve for \begin{align*}r\end{align*}? Describe the steps you would use in words. You do not have to solve the problem!
Supplementary Angles
Two angles are supplementary if their measures sum to \begin{align*}180^\circ\end{align*}.
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 \begin{align*}180^\circ\end{align*}. You can use this information just as you did with complementary angles to solve different types of problems.
- The measures of supplementary angles add up to ________________.
Example 3
The two angles below are supplementary. If \begin{align*}m\angle MNO=78^\circ\end{align*}, what is \begin{align*}m\angle PQR\end{align*}?
Use a variable for the unknown angle measure and then solve for the variable. In this case, let's substitute \begin{align*}y\end{align*} for \begin{align*}m\angle PQR\end{align*}.
If the angles are supplementary, then their measures add up to ____________.
\begin{align*}&m\angle MNO + m\angle PQR = 180^\circ\\ &\qquad \qquad \qquad \ \ 78 + y = 180\\ &\qquad \qquad \quad \ - 78 \qquad \ -78\\ &\qquad \qquad \qquad \qquad \ \ y = 102\end{align*}
So, the measure of \begin{align*}y = 102\end{align*} and thus \begin{align*}m\angle PQR = 102^\circ\end{align*}.
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, \begin{align*}\angle PQR\end{align*} and \begin{align*}\angle RQS\end{align*} are adjacent:
However, \begin{align*}\angle PQR\end{align*} and \begin{align*}\angle PQS\end{align*} are not adjacent since they overlap (i.e. they share common points in the interior of the angle).
- ____________________________ angles are next to each other: they share the same vertex and one side.
Adjacent is a word meaning “next to.”
Things that are adjacent are usually touching, and they share a border.
Oregon, Nevada, and Arizona are adjacent to California because they share a border with California.
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, \begin{align*}\angle MNP\end{align*} and \begin{align*}\angle PNO\end{align*} are a linear pair. Note that \begin{align*}\overleftrightarrow{MO}\end{align*} is a line.
- Linear pairs are angles that are next to each other along 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 4
The two angles below form a linear pair. What is the value of each angle?
We just learned that linear pairs are ____________________________________, so we know that they add up to \begin{align*}180^\circ\end{align*}.
The best way to solve this problem is to set up an equation where the two angle expressions sum to \begin{align*}180^\circ\end{align*}. Then solve the equation for \begin{align*}q\end{align*}.
\begin{align*}m\angle UTS{\;\;\;\;\;} + {\;\;\;\;\;\;\;} m\angle STV{\;\;\;\;\;\;} &= 180^\circ\\ ( \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;} ) + ( \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;} ) &= 180^\circ\end{align*}
To find the measure of each angle, you must substitute the value for \begin{align*}q\end{align*} back into the original expressions to find the value of each angle.
\begin{align*}(3q)+ ( 15q+18 )&= \ 180\\ 18q+18& = \ 180\\ - 18 & \quad -18\\ 18q & = \ 162\\ q & = \ 9\end{align*}
The value of \begin{align*}q\end{align*} is 9. Now substitute this value back into both angle expressions to find the measures of the two angles in the diagram:
\begin{align*}&3q && 15q+18\\ &3(9) && 15(9)+ 18\\ &27 && 135 +18\\ &&&153\end{align*}
\begin{align*}m\angle UTS = 27^\circ\end{align*} and \begin{align*}m\angle STV = 53^\circ\end{align*}. You can check to make sure these numbers are accurate by verifying that they are supplementary (add up to \begin{align*}180^\circ\end{align*}):
\begin{align*}27 + 53 = 180\end{align*}
Since these two angle measures sum to \begin{align*}180^\circ\end{align*}, they are supplementary.
Reading Check:
1. Fill in the blank:
Linear pairs add up to \begin{align*}180^\circ\end{align*}. In other words, they are ________________________.
2. Find the measure of angle \begin{align*}\angle JKL\end{align*} in the picture below:
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, \begin{align*}\angle 1\end{align*} and \begin{align*}\angle 3\end{align*} are vertical angles. Also, \begin{align*}\angle 4\end{align*} and \begin{align*}\angle 2\end{align*} are vertical angles.
Vertical angles are non-______________________________, which means they are not next to each other.
Vertical angles are formed by _______________________________ lines, and as you can see in the diagram below, they are always directly across from each other at the intersection:
Suppose that you know \begin{align*}m\angle 1 = 100^\circ\end{align*}. You can use that information to find the measurement of all of the other angles.
For example, \begin{align*}\angle 1\end{align*} and \begin{align*}\angle 2\end{align*} must be supplementary since they are a linear pair.
So, to find \begin{align*}m\angle 2\end{align*}, subtract \begin{align*}100^\circ\end{align*} from \begin{align*}180^\circ\end{align*}:
\begin{align*}m\angle 1 + m\angle 2 & \ = \ 180^\circ\\ 100^\circ + m\angle 2 &\ = \ 180^\circ\\ - 100^\circ{\;\;\;\;\;\;\;\;\;\;} & \quad -100^\circ\\ m\angle 2 & \ = \ 80^\circ\end{align*}
So \begin{align*}\angle 2\end{align*} measures \begin{align*}80^\circ\end{align*}. Knowing that angles \begin{align*}\angle 2\end{align*} and \begin{align*}\angle 3\end{align*} are also supplementary means that \begin{align*}m\angle 3 = 100^\circ\end{align*}, since the sum of \begin{align*}100^\circ\end{align*} and \begin{align*}80^\circ\end{align*} is \begin{align*}180^\circ\end{align*}.
If angle \begin{align*}\angle 3\end{align*} measures \begin{align*}100^\circ\end{align*}, then the measure of angle \begin{align*}\angle 4\end{align*} must be \begin{align*}80^\circ\end{align*}, since \begin{align*}\angle 3\end{align*} and \begin{align*}\angle 4\end{align*} are also supplementary.
Notice that angles \begin{align*}\angle 1\end{align*} and \begin{align*}\angle 3\end{align*} are congruent \begin{align*}(100^\circ)\end{align*} and \begin{align*}\angle 2\end{align*} and \begin{align*}\angle 4\end{align*} are congruent \begin{align*}(80^\circ)\end{align*}.
Vertical Angles Theorem
The Vertical Angles Theorem states that if two angles are vertical angles then they are congruent.
- Vertical angles are _____________________________ to each other.
Introduction to Proof: Proving the Vertical Angle Theorem
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 \begin{align*}\angle 1\end{align*}.
Here is proof that vertical angles will always be congruent:
Since \begin{align*}\angle 1\end{align*} and \begin{align*}\angle 2\end{align*} form a linear pair, we know that they are supplementary:
\begin{align*}m\angle 1 + m\angle 2 = 180^\circ\end{align*}
For the same reason, \begin{align*}\angle 2\end{align*} and \begin{align*}\angle 3\end{align*} are supplementary: \begin{align*}m\angle 2 + m\angle 3 = 180^\circ\end{align*}
Using a substitution (they both \begin{align*}= 180^\circ\end{align*}), we can write \begin{align*}m\angle 1 + m\angle 2 = m\angle 2 + m\angle 3\end{align*}.
Finally, subtracting \begin{align*}m\angle 2\end{align*} on both sides gives us \begin{align*}m\angle 1 = m\angle 3\end{align*}.
Or, by the definition of congruent angles: \begin{align*}\angle 1 \cong \angle 3\end{align*}.
Reading Check:
1. Fill in the blank: The reason that vertical angles are congruent is that each pair of adjacent angles is a ___________________________ pair.
2. Find the measure of angle \begin{align*}\angle CAT\end{align*} in the picture below.
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