<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />

1.5: Angle Pairs

Difficulty Level: At Grade Created by: CK-12

Learning Objectives

• Recognize complementary angles supplementary angles, linear pairs, and vertical angles.
• Apply the Linear Pair Postulate and the Vertical Angles Theorem.

Review Queue

1. Find x\begin{align*}x\end{align*}.
2. Find y\begin{align*}y\end{align*}.
3. Find z\begin{align*}z\end{align*}.

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 45 NW\begin{align*}45^\circ \ NW\end{align*}, would be straight out along that northwest line.

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 90\begin{align*}90^\circ\end{align*}.

Complementary angles do not have to be:

• congruent
• next to each other

Example 1: The two angles below are complementary. mGHI=x\begin{align*}m\angle GHI = x\end{align*}. What is x\begin{align*}x\end{align*}?

Solution: Because the two angles are complementary, they add up to 90\begin{align*}90^\circ\end{align*}. Make an equation.

x+34x=90=56\begin{align*}x + 34^\circ & = 90^\circ\\ x & = 56^\circ\end{align*}

Example 2: The two angles below are complementary. Find the measure of each angle.

Solution: The two angles add up to 90\begin{align*}90^\circ\end{align*}. Make an equation.

8r+9+7r+615r+1515rr=90=90=75=5\begin{align*}8r + 9^\circ + 7r + 6^\circ & = 90^\circ\\ 15r + 15^\circ & = 90^\circ\\ 15r & = 75^\circ\\ r & = 5^\circ\end{align*}

However, you need to find each angle. Plug r\begin{align*}r\end{align*} back into each expression.

mGHImJKL=8(5)+9=49=7(5)+6=41\begin{align*}m \angle GHI & = 8(5^\circ) + 9^\circ = 49^\circ\\ m\angle JKL & = 7(5^\circ) + 6^\circ = 41^\circ\end{align*}

Supplementary Angles

Supplementary: Two angles that add up to 180\begin{align*}180^\circ\end{align*}.

Supplementary angles do not have to be:

• congruent
• next to each other

Example 3: The two angles below are supplementary. If mMNO=78\begin{align*}m\angle MNO = 78^\circ\end{align*} what is mPQR\begin{align*}m\angle PQR\end{align*}?

Solution: Set up an equation. However, instead of equaling 90\begin{align*}90^\circ\end{align*}, now it is 180\begin{align*}180^\circ\end{align*}.

78+mPQRmPQR=180=102\begin{align*}78^\circ + m\angle PQR & = 180^\circ\\ m\angle PQR & = 102^\circ\end{align*}

Example 4: What is the measure of two congruent, supplementary angles?

Solution: Supplementary angles add up to 180\begin{align*}180^\circ\end{align*}. Congruent angles have the same measure. So, 180÷2=90\begin{align*}180^\circ \div 2 = 90^\circ\end{align*}, which means two congruent, supplementary angles are right angles, or 90\begin{align*}90^\circ\end{align*}.

Linear Pairs

Adjacent Angles: Two angles that have the same vertex, share a side, and do not overlap.

PSQ\begin{align*}\angle PSQ\end{align*} and QSR\begin{align*}\angle QSR\end{align*} are adjacent.

PQR\begin{align*}\angle PQR\end{align*} and PQS\begin{align*}\angle PQS\end{align*} are NOT adjacent because they overlap.

Linear Pair: Two angles that are adjacent and the non-common sides form a straight line.

PSQ\begin{align*}\angle PSQ\end{align*} and QSR\begin{align*}\angle QSR\end{align*} are a linear pair.

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 180\begin{align*}180^\circ\end{align*}.

(7q46)+(3q+6)10q4010qq=180=180=220=22\begin{align*}(7q-46)^\circ + (3q+6)^\circ &= 180^\circ\\ 10q - 40^\circ &= 180^\circ\\ 10q & = 220^\circ\\ q & = 22^\circ\end{align*}

Plug in q\begin{align*}q\end{align*} to get the measure of each angle. mABD=7(22)46=108 mDBC=180108=72\begin{align*}m\angle ABD = 7(22^\circ) - 46^\circ = 108^\circ \ m\angle DBC = 180^\circ - 108^\circ = 72^\circ\end{align*}

Example 6: Are CDA\begin{align*}\angle CDA\end{align*} and DAB\begin{align*}\angle DAB\end{align*} a linear pair? Are they supplementary?

Solution: The two angles are not a linear pair because they do not have the same vertex. They are supplementary, 120+60=180\begin{align*}120^\circ + 60^\circ = 180^\circ\end{align*}.

Vertical Angles

Vertical Angles: Two non-adjacent angles formed by intersecting lines.

1\begin{align*}\angle 1\end{align*} and 3\begin{align*}\angle 3\end{align*} are vertical angles

2\begin{align*}\angle 2\end{align*} and 4\begin{align*}\angle 4\end{align*} are vertical angles

These angles are labeled with numbers. You can tell that these are labels because they do not have a degree symbol.

Investigation 1-6: Vertical Angle Relationships

1. Draw two intersecting lines on your paper. Label the four angles created 1, 2, 3\begin{align*}\angle 1, \ \angle 2, \ \angle 3\end{align*}, and 4\begin{align*}\angle 4\end{align*}, just like the picture above.
2. Use your protractor to find m1\begin{align*}m\angle 1\end{align*}.
3. What is the angle relationship between 1\begin{align*}\angle 1\end{align*} and 2\begin{align*}\angle 2\end{align*} called? Find m2\begin{align*}m\angle 2\end{align*}.
4. What is the angle relationship between 1\begin{align*}\angle 1\end{align*} and 4\begin{align*}\angle 4\end{align*} called? Find m4\begin{align*}m\angle 4\end{align*}.
5. What is the angle relationship between 2\begin{align*}\angle 2\end{align*} and 3\begin{align*}\angle 3\end{align*} called? Find m3\begin{align*}m\angle 3\end{align*}.
6. Are any angles congruent? If so, write them down.

From this investigation, you should find that 13\begin{align*}\angle 1 \cong \angle 3\end{align*} and 24\begin{align*}\angle 2 \cong \angle 4\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:

1 and 2 are a linear pair m1+m22 and 3 are a linear pair m2+m33 and 4 are a linear pair m3+m4=180Equation 1=180Equation 2=180Equation 3\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 =180\begin{align*}= 180^\circ\end{align*}, so Equation 1 = Equation 2 and Equation 2 = Equation 3.

m1+m2=m2+m3andm2+m3=m3+m4\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

m1=m3andm2=m4\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, 13\begin{align*}\angle 1 \cong \angle 3\end{align*} and 24\begin{align*}\angle 2 \cong \angle 4\end{align*} too.

Example 7: Find m1\begin{align*}m\angle 1\end{align*} and m2\begin{align*}m\angle 2\end{align*}.

Solution: 1\begin{align*}\angle 1\end{align*} is vertical angles with 18\begin{align*}18^\circ\end{align*}, so m1=18\begin{align*}m\angle 1 = 18^\circ\end{align*}.

2\begin{align*}\angle 2\end{align*} is a linear pair with 1\begin{align*}\angle 1\end{align*} or 18\begin{align*}18^\circ\end{align*}, so 18+m2=180\begin{align*}18^\circ + m\angle 2 = 180^\circ\end{align*}.

m2=18018=162\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 22.5,180÷8\begin{align*}22.5^\circ, 180^\circ \div 8\end{align*}. Directions that are opposite each other have the same angle measure, but of course, a different direction. All of the green directions have the same angle measure, 22.5\begin{align*}22.5^\circ\end{align*}, and the purple have the same angle measure, 45\begin{align*}45^\circ\end{align*}. N, S, E\begin{align*}N, \ S, \ E\end{align*} and W\begin{align*}W\end{align*} all have different measures, even though they are all 90\begin{align*}90^\circ\end{align*} apart.

Review Questions

• Questions 1 and 2 are similar to Examples 1, 2, and 3.
• Questions 3-8 are similar to Examples 3, 4, 6 and 7.
• Questions 9-16 use the definitions, postulates and theorems from this section.
• Questions 17-25 are similar to Example 5.
1. Find the measure of an angle that is complementary to ABC\begin{align*}\angle ABC\end{align*} if mABC\begin{align*}m\angle ABC\end{align*} is
1. 45\begin{align*}45^\circ\end{align*}
2. 82\begin{align*}82^\circ\end{align*}
3. 19\begin{align*}19^\circ\end{align*}
4. z\begin{align*}z^\circ\end{align*}
2. Find the measure of an angle that is supplementary to ABC\begin{align*}\angle ABC\end{align*} if mABC\begin{align*}m\angle ABC\end{align*} is
1. 45\begin{align*}45^\circ\end{align*}
2. 118\begin{align*}118^\circ\end{align*}
3. 32\begin{align*}32^\circ\end{align*}
4. x\begin{align*}x^\circ\end{align*}

Use the diagram below for exercises 3-7. Note that NK¯¯¯¯¯¯¯¯¯IL\begin{align*}\overline{NK} \perp \overleftrightarrow{IL}\end{align*}.

1. Name one pair of vertical angles.
2. Name one linear pair of angles.
3. Name two complementary angles.
4. Name two supplementary angles.
1. What is:
1. mINL\begin{align*}m\angle INL\end{align*}
2. mLNK\begin{align*}m\angle LNK\end{align*}
2. If mINJ=63\begin{align*}m\angle INJ = 63^\circ\end{align*}, find:
1. mJNL\begin{align*}m\angle JNL\end{align*}
2. mKNJ\begin{align*}m\angle KNJ\end{align*}
3. mMNL\begin{align*}m\angle MNL\end{align*}
4. mMNI\begin{align*}m\angle MNI\end{align*}

For 9-16, determine if the statement is true or false.

1. Vertical angles are congruent.
2. Linear pairs are congruent.
3. Complementary angles add up to 180\begin{align*}180^\circ\end{align*}.
4. Supplementary angles add up to 180\begin{align*}180^\circ\end{align*}
5. Adjacent angles share a vertex.
7. Complementary angles are always 45\begin{align*}45^\circ\end{align*}.
8. Vertical angles have the same vertex.

For 17-25, find the value of x\begin{align*}x\end{align*} or y\begin{align*}y\end{align*}.

1. Find x\begin{align*}x\end{align*}.
2. Find y\begin{align*}y\end{align*}.

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

Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes

Show Hide Details
Description
Tags:
Subjects: