Introduction

Two angles are congruent if their measures are equal.

For example, if

m\angle X =50^\circ \quad \text{and} \ \quad m\angle Y=50^\circ,

then \angle X and \angle Y are congruent. We can express this as

\angle X \cong \angle Y.

However, if

m\angle X =50^\circ \quad \text{and} \ \quad m\angle Z = 110^\circ,

then \angle X and \angle Z are not congruent, and we write

\angle X \ncong \angle Z.

We represent congruent angles by marking the arcs with the same symbol. For example, three pairs of congruent angles are show below:

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Example: Identifying Congruent Angles Given Measures of Angles

From the information given below, find all of the angles that are congruent. \begin{align} m \angle A &= 10^\circ, \\[3pt] m \angle B &= 40^\circ, \\[3pt] m \angle C &= 80^\circ, \\[3pt] m \angle D &= 90^\circ-m \angle C, \\[3pt] m \angle E &= m \angle B. \end{align}

EXPLANATION

First, let's find the measures of \angle D and \angle E\mathbin{:}

\begin{align} m \angle D &= 90^\circ-m \angle C \\ &= 90^\circ-80^\circ \\ &= 10^\circ \\ \\ m \angle E &= m \angle B \\ & = 40^\circ \end{align}

So, we have:

\begin{align} m \angle A &= 10^\circ \\[3pt] m \angle B &= 40^\circ \\[3pt] m \angle C &= 80^\circ \\[3pt] m \angle D &= 10^\circ \\[3pt] m \angle E &= 40^\circ \end{align}

We have that m \angle A = m \angle D and m \angle B = m \angle E.

Therefore, \angle A \cong \angle D and \angle B \cong \angle E.

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Practice: Identifying Congruent Angles Given Measures of Angles

Which of the angles shown above are congruent?

a
 No congruent angles
b
 $\angle 1 \cong \angle 2 \cong \angle 3$
c
 $\angle 1 \cong \angle 2$
d
 $\angle 2 \cong \angle 3$
e
 $\angle 1 \cong \angle 3$
Practice: Identifying Congruent Angles Given Measures of Angles

From the information given below, find all of the angles that are congruent. \begin{align} m \angle A &= 60^\circ, \\[3pt] m \angle B &= 30^\circ, \\[3pt] m \angle C &= 20^\circ, \\[3pt] m \angle D &= m \angle A, \\[3pt] m \angle E &= 90^\circ-m \angle B, \\[3pt] m \angle F &= 180^\circ-m \angle C. \\ \end{align}

a
 $\angle A \cong \angle E$ and $\angle D \cong \angle F$
b
 $\angle A \cong \angle D$ and $\angle B \cong \angle E$
c
 $\angle A \cong \angle D$ and $\angle E \cong \angle F$
d
 $\angle C \cong \angle E\cong \angle F$
e
 $\angle A \cong \angle D\cong \angle E$
Example: Using Markings to Denote Congruent Angles

Which of the following statements are true regarding the angles shown below?

  1. \angle AOB \cong \angle COD
  2. \angle COD \cong \angle BOC
  3. m\angle BOC = m\angle DOE
EXPLANATION

From the diagram, we see that

  • \angle AOB & \angle COD are each marked with a single hash mark " | ," while

  • \angle BOC & \angle DOE are each marked with a double hash mark " ||. "

Therefore, \begin{align*} \angle AOB &\cong \angle COD \\ \angle BOC &\cong \angle DOE. \\ \end{align*}

Consequently, we have

\begin{align*} m\angle AOB &= m\angle COD \\ m\angle BOC &= m\angle DOE. \\ \end{align*}

So statements I and III are true, while statement II is false.

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Practice: Using Markings to Denote Congruent Angles

Which of the following statements are true regarding the angles shown above?

  1. $\angle AOB \cong \angle BOC$
  2. $m\angle BOC = m\angle EOF$
  3. $\angle DOE \cong \angle COD$
a
 I and III only
b
 I and II only
c
 II and III only
d
 II only
e
 I only
Practice: Using Markings to Denote Congruent Angles

Which of the following statements are true regarding the angles shown above?

  1. $m\angle EOA = m\angle COD$
  2. $m\angle AOB = m\angle COD$
  3. $\angle EOA \cong \angle BOC$
a
 I only
b
 II only
c
 III only
d
 I, II and III
e
 II and III only
Example: Solving Problems Involving Congruent Angles

In the figure above, m \angle CBD = 5x - 5^\circ and m \angle ABD =2x + 10^\circ . If \angle CBD \cong \angle ABD , find m \angle ABC.

EXPLANATION

Since \angle CBD \cong \angle ABD , we have \begin{align} m \angle CBD & = m \angle ABD\\[5pt] 5x - 5^\circ & = 2x + 10^\circ\\[5pt] 5x & = 2x + 15^\circ\\[5pt] 3x & = 15^\circ\\[5pt] x & = 5^\circ. \end{align}

Therefore, the measure of \angle CBD is m \angle CBD = 5x - 5^\circ = 5 \cdot 5^\circ - 5^\circ = 20^\circ and the measure of \angle ABD is also m \angle ABD =2x + 10^\circ = 2 \cdot 5^\circ + 10^\circ = 20^\circ .

So, by the angle addition postulate, \begin{align} m \angle ABC & = m \angle CBD + m \angle ABD\\ & = 20^\circ + 20^\circ \\ & = 40^\circ. \end{align}

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Practice: Solving Problems Involving Congruent Angles

In the figure above, $m \angle CBD = 3x + 5^\circ$ and $m \angle ABD = 6x - 19^\circ.$ If $\angle CBD \cong \angle ABD$, find $m \angle ABC.$

a
 $54^\circ$
b
 $60^\circ$
c
 $58^\circ$
d
 $61^\circ$
e
 $62^\circ$
Practice: Solving Problems Involving Congruent Angles

In the figure above, $\angle ABC$ is an angle and $D$ is a point in the interior of $\angle ABC$ such that $m \angle CBD = 4x + 3^\circ.$ If $m \angle ABC = 110^\circ$ and $\angle CBD \cong \angle ABD$, find the value of $x.$

a
 $17^\circ$
b
 $14^\circ$
c
 $10^\circ$
d
 $13^\circ$
e
 $16^\circ$
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