### Congruent Angles and Bisectors

When two geometric figures have the same shape and size (or the same angle measure in the case of angles) they are said to be **congruent**.

Label It |
Say It |
---|---|

\begin{align*}\angle ABC \cong \angle DEF\end{align*} | Angle \begin{align*}ABC\end{align*} is congruent to angle \begin{align*}DEF\end{align*}. |

If two angles are congruent, then they are also equal. To label equal angles we use **angle markings**, as shown below:

An **angle bisector** is a line, or a portion of a line, that divides an angle into two congruent angles, each having a measure exactly half of the original angle. Every angle has exactly one angle bisector.

In the picture above, \begin{align*}\overline{BD}\end{align*} is the angle bisector of \begin{align*}\angle ABC\end{align*}, so \begin{align*}\angle ABD \cong \angle DBC\end{align*} and \begin{align*}m\angle ABD = \frac{1}{2} m \angle ABC\end{align*}.

What if you were told that a line segment divides an angle in half? How would you find the measures of the two new angles formed by that segment?

### Examples

For Examples 1 and 2, copy the figure below and label it with the following information:

#### Example 1

\begin{align*}\angle A \cong \angle C\end{align*}

You should have corresponding markings on \begin{align*}\angle A \end{align*} and \begin{align*}\angle C\end{align*}.

#### Example 2

\begin{align*}\angle B \cong \angle D\end{align*}

You should have corresponding markings on \begin{align*}\angle B \end{align*} and \begin{align*}\angle D\end{align*} (that look different from the markings you made in Example 1).

#### Example 3

Write all equal angle statements.

\begin{align*}m\angle ADB &= m \angle BDC = m \angle FDE = 45^\circ\\ m \angle ADF &= m \angle ADC = 90^\circ\end{align*}

#### Example 4

What is the measure of each angle?

From the picture, we see that the angles are equal.

Set the angles equal to each other and solve.

\begin{align*}(5x + 7)^\circ &= (3x + 23)^\circ\\ (2x)^\circ &= 16^\circ\\ x &= 8\end{align*}

To find the measure of \begin{align*}\angle ABC\end{align*}, plug in \begin{align*}x = 8\end{align*} to

\begin{align*}(5x + 7)^\circ \rightarrow (5(8) + 7)^\circ = (40 + 7)^\circ = 47^\circ\end{align*}.

Because \begin{align*}m \angle ABC = m \angle XYZ, \ m \angle XYZ = 47^\circ\end{align*} too.

#### Example 5

Is \begin{align*}\overline{OP}\end{align*} the angle bisector of \begin{align*}\angle SOT\end{align*}?

Yes, \begin{align*}\overline{OP}\end{align*} is the angle bisector of \begin{align*}\angle SOT\end{align*} from the markings in the picture.

### Review

For 1-4, use the following picture to answer the questions.

- What is the angle bisector of \begin{align*}\angle TPR\end{align*}?
- What is \begin{align*}m\angle QPR\end{align*}?
- What is \begin{align*}m\angle TPS\end{align*}?
- What is \begin{align*}m\angle QPV\end{align*}?

For 5-6, use algebra to determine the value of variable in each problem.

For 7-10, decide if the statement is true or false.

- Every angle has exactly one angle bisector.
- Any marking on an angle means that the angle is \begin{align*}90^\circ\end{align*}.
- An angle bisector divides an angle into three congruent angles.
- Congruent angles have the same measure.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 1.3.