1.5: Angle Pairs
Learning Objectives
 Recognize complementary angles supplementary angles, linear pairs, and vertical angles.
 Apply the Linear Pair Postulate and the Vertical Angles Theorem.
Review Queue
 Find
x .  Find
y .  Find
z .
Know What? A compass (as seen to the right) is used to determine the direction a person is traveling. The angles between each direction are very important because they enable someone to be more specific with their direction. A direction of
What headings have the same angle measure? What is the angle measure between each compass line?
Complementary Angles
Complementary: Two angles that add up to
Complementary angles do not have to be:
 congruent
 next to each other
Example 1: The two angles below are complementary.
Solution: Because the two angles are complementary, they add up to
Example 2: The two angles below are complementary. Find the measure of each angle.
Solution: The two angles add up to
However, you need to find each angle. Plug
Supplementary Angles
Supplementary: Two angles that add up to
Supplementary angles do not have to be:
 congruent
 next to each other
Example 3: The two angles below are supplementary. If
Solution: Set up an equation. However, instead of equaling
Example 4: What is the measure of two congruent, supplementary angles?
Solution: Supplementary angles add up to
Linear Pairs
Adjacent Angles: Two angles that have the same vertex, share a side, and do not overlap.
Linear Pair: Two angles that are adjacent and the noncommon sides form a straight line.
Linear Pair Postulate: If two angles are a linear pair, then they are supplementary.
Example 5: Algebra Connection What is the measure of each angle?
Solution: These two angles are a linear pair, so they add up to
Plug in
Example 6: Are
Solution: The two angles are not a linear pair because they do not have the same vertex. They are supplementary,
Vertical Angles
Vertical Angles: Two nonadjacent angles formed by intersecting lines.
These angles are labeled with numbers. You can tell that these are labels because they do not have a degree symbol.
Investigation 16: Vertical Angle Relationships
 Draw two intersecting lines on your paper. Label the four angles created
∠1, ∠2, ∠3 , and∠4 , just like the picture above.  Use your protractor to find
m∠1 .  What is the angle relationship between
∠1 and∠2 called? Findm∠2 .  What is the angle relationship between
∠1 and∠4 called? Findm∠4 .  What is the angle relationship between
∠2 and∠3 called? Find \begin{align*}m\angle 3\end{align*}m∠3 .  Are any angles congruent? If so, write them down.
From this investigation, you should find that \begin{align*}\angle 1 \cong \angle 3\end{align*}
Vertical Angles Theorem: If two angles are vertical angles, then they are congruent.
We can prove the Vertical Angles Theorem using the same process we used in the investigation. We will not use any specific values for the angles.
From the picture above:
\begin{align*}\angle 1 \ \text{and} \ \angle 2 \ \text{are a linear pair} \ \rightarrow m\angle 1 + m\angle 2 & = 180^\circ \qquad \text{Equation} \ 1\\
\angle 2 \ \text{and} \ \angle 3 \ \text{are a linear pair} \ \rightarrow m\angle 2 + m\angle 3 & = 180^\circ \qquad \text{Equation} \ 2\\
\angle 3 \ \text{and} \ \angle 4 \ \text{are a linear pair} \ \rightarrow m\angle 3 + m\angle 4 & = 180^\circ \qquad \text{Equation} \ 3\end{align*}
All of the equations \begin{align*}= 180^\circ\end{align*}
\begin{align*}m\angle 1 + m\angle 2 = m\angle 2 + m\angle 3 \qquad \text{and} \qquad m\angle 2 + m\angle 3 = m\angle 3 + m\angle 4\end{align*}
Cancel out the like terms
\begin{align*}m\angle 1 = m\angle 3 \qquad \text{and} \qquad m\angle 2 = m\angle 4\end{align*}
Recall that anytime the measures of two angles are equal, the angles are also congruent. So, \begin{align*}\angle 1 \cong \angle 3\end{align*}
Example 7: Find \begin{align*}m\angle 1\end{align*}
Solution: \begin{align*}\angle 1\end{align*}
\begin{align*}\angle 2\end{align*}
\begin{align*}m\angle 2 = 180^\circ  18^\circ = 162^\circ\end{align*}
Know What? Revisited The compass has several vertical angles and all of the smaller angles are \begin{align*}22.5^\circ, 180^\circ \div 8\end{align*}
Review Questions
 Questions 1 and 2 are similar to Examples 1, 2, and 3.
 Questions 38 are similar to Examples 3, 4, 6 and 7.
 Questions 916 use the definitions, postulates and theorems from this section.
 Questions 1725 are similar to Example 5.
 Find the measure of an angle that is complementary to \begin{align*}\angle ABC\end{align*}
∠ABC if \begin{align*}m\angle ABC\end{align*}m∠ABC is
\begin{align*}45^\circ\end{align*}
45∘ 
\begin{align*}82^\circ\end{align*}
82∘ 
\begin{align*}19^\circ\end{align*}
19∘ 
\begin{align*}z^\circ\end{align*}
z∘

\begin{align*}45^\circ\end{align*}
 Find the measure of an angle that is supplementary to \begin{align*}\angle ABC\end{align*}
∠ABC if \begin{align*}m\angle ABC\end{align*}m∠ABC is
\begin{align*}45^\circ\end{align*}
45∘ 
\begin{align*}118^\circ\end{align*}
118∘ 
\begin{align*}32^\circ\end{align*}
32∘ 
\begin{align*}x^\circ\end{align*}
x∘

\begin{align*}45^\circ\end{align*}
Use the diagram below for exercises 37. Note that \begin{align*}\overline{NK} \perp \overleftrightarrow{IL}\end{align*}
 Name one pair of vertical angles.
 Name one linear pair of angles.
 Name two complementary angles.
 Name two supplementary angles.
 What is:

\begin{align*}m\angle INL\end{align*}
m∠INL 
\begin{align*}m\angle LNK\end{align*}
m∠LNK

\begin{align*}m\angle INL\end{align*}
 If \begin{align*}m\angle INJ = 63^\circ\end{align*}
m∠INJ=63∘ , find:
\begin{align*}m\angle JNL\end{align*}
m∠JNL 
\begin{align*}m\angle KNJ\end{align*}
m∠KNJ 
\begin{align*}m\angle MNL\end{align*}
m∠MNL 
\begin{align*}m\angle MNI\end{align*}
m∠MNI

\begin{align*}m\angle JNL\end{align*}
For 916, determine if the statement is true or false.
 Vertical angles are congruent.
 Linear pairs are congruent.
 Complementary angles add up to \begin{align*}180^\circ\end{align*}
180∘ .  Supplementary angles add up to \begin{align*}180^\circ\end{align*}
180∘  Adjacent angles share a vertex.
 Adjacent angles overlap.
 Complementary angles are always \begin{align*}45^\circ\end{align*}
45∘ .  Vertical angles have the same vertex.
For 1725, find the value of \begin{align*}x\end{align*}
 Find \begin{align*}x\end{align*}
x .  Find \begin{align*}y\end{align*}
y .
Review Queue Answers

\begin{align*}x+26 = 3x8\!\\
{\;} \quad \ 34 = 2x\!\\
{\;} \quad \ 17 = x\end{align*}
x+26=3x−8 34=2x 17=x 
\begin{align*}(7y+6)^\circ = 90^\circ\!\\
{\;} \qquad \ 7y = 84^\circ\!\\
{\;} \qquad \ \ y = 12^\circ\end{align*}
(7y+6)∘=90∘ 7y=84∘ y=12∘ 
\begin{align*}z+ 15 = 5z + 9\!\\
{\;} \quad \ \ 6 = 4z\!\\
{\;} \quad 1.5 = z\end{align*}
z+15=5z+9 6=4z1.5=z
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