Introduction

When two angles share a common side, they are called adjacent angles. When two angles are adjacent, we can add their measures to find the measure of the larger angle.

For example, consider the diagram below.

The angles \angle PSR and \angle RST are adjacent because they both have the side \overrightarrow{SR} in common.

Since the angles \angle PSR and \angle RST are adjacent, we can find the measure of the combined angle \angle PST by adding the two angles:

\begin{align*} m\angle PST &=\underbrace{m \angle PSR}_{\color[rgb]{0.902, 0.624, 0}{\textrm{orange angle}}} + \underbrace{m \angle RST}_{\color[rgb]{0.173,0.447,0.682}{\textrm{blue angle}}}\\[5pt] &=95^\circ + 45^\circ\\[5pt] &=140^\circ \end{align*}

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Example: Finding the Measure of a Combined Angle

In the figure below, determine m\angle{POR}.

EXPLANATION

First, notice that \angle{POQ} and \angle{QOR} are adjacent because they share the same side \overrightarrow{OQ}.

We can add the measures of two adjacent angles to find the measure of the combined angle.

Therefore: \begin{align*} m \angle{POR} & = m\angle{POQ} + m\angle{QOR}\\[5pt] & = 68^{\circ} + 37^{\circ}\\[5pt] & = 105^{\circ} \end{align*}

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Practice: Finding the Measure of a Combined Angle

What is the measure of $\widehat{COE}?$

a
 $105^{\circ}$
b
 $110^{\circ}$
c
 $100^{\circ}$
d
 $102^{\circ}$
e
 $114^{\circ}$
Practice: Finding the Measure of a Combined Angle

In the figure above, find $m\angle AOC.$

a
 ${75}^{\circ}$
b
 ${74}^{\circ}$
c
 ${82}^{\circ}$
d
 ${78}^{\circ}$
e
 ${80}^{\circ}$
Example: Finding the Measure of a Smaller Angle

The measure of the angle \angle BAD is {90}^{\circ}. Find the value of x.

EXPLANATION

Here, the letter x represents the measure of the unknown angle \angle CAD.

Note the following:

  • The angles \angle BAC and \angle CAD are adjacent because they share the side \overrightarrow{AC}.

  • So, the measure of the large angle (\angle BAD) must be the sum of the measures of the two smaller angles (\angle BAC and \angle CAD).

  • This means that the measure of \angle CAD must be the difference between the large angle (\angle BAD) and the smaller known angle (\angle BAC).

Therefore,

x = 90^\circ - 35^\circ = 55^\circ.

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Practice: Finding the Measure of a Smaller Angle

In the diagram above, $m\angle BAD = {90}^{\circ}.$ What equation could be used to find the value of $x?$

a
 $x = 90^\circ + 40^\circ$
b
 $x = 40^\circ \times 90^\circ$
c
 $x = 40^\circ + 90^\circ$
d
 $x = 40^\circ - 90^\circ$
e
 $x = 90^\circ - 40^\circ$
Practice: Finding the Measure of a Smaller Angle

The measure of the angle $\angle BAD$ is ${80}^{\circ}.$ Find the value of $x.$

a
 $30^{\circ}$
b
 $110^{\circ}$
c
 $50^{\circ}$
d
 $40^{\circ}$
e
 $60^{\circ}$
Extending the Sums of Angles Rule

We can extend our rule relating sums of adjacent angles to more than two angles.

For example, let's consider the angles shown in the diagram below.

According to our rule, the measure of the reflex angle \angle QST is

\begin{align*} m\angle QST &= \underbrace{m \angle QSP}_{\color[rgb]{0.29, 0.6, 0.47}{\textrm{green angle}}} + \underbrace{m \angle PSR}_{\color[rgb]{0.902, 0.624, 0}{\textrm{orange angle}}} + \underbrace{m \angle RST}_{\color[rgb]{0.173,0.447,0.682}{\textrm{blue angle}}}\\[5pt] &=100^\circ + 95^\circ + 45^\circ\\[5pt] &=240^\circ. \end{align*}

We can also use this rule to determine the size of a smaller angle when we know a larger one. Let's see an example.

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Example: Finding the Measure of an Unknown Angle Using the Extended Rule

In the figure below, m \angle BAE = {110}^{\circ}. Find the value of x.

EXPLANATION

Here, the letter x represents the measure of the unknown angle \angle CAD.

The measure of \angle CAD must be the difference between the large angle (\angle BAE) and the two smaller known angles (\angle BAC and \angle DAE).

Therefore, x = 110^\circ - 40^\circ - 20^\circ = 50^\circ.

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Practice: Finding the Measure of an Unknown Angle Using the Extended Rule

In the figure above, $m \angle BAE = {135}^{\circ}.$ Find the value of $x.$

a
 ${65}^{\circ}$
b
 ${55}^{\circ}$
c
 ${60}^{\circ}$
d
 ${58}^{\circ}$
e
 ${63}^{\circ}$
Practice: Finding the Measure of an Unknown Angle Using the Extended Rule

In the figure above, $m \angle BAE = {160}^{\circ}$ and $m \angle BAD = {110}^{\circ}.$ Find $m \angle CAE.$

a
 ${100}^{\circ}$
b
 ${145}^{\circ}$
c
 ${110}^{\circ}$
d
 ${130}^{\circ}$
e
 ${150}^{\circ}$
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