### Congruent Angles and Angle Bisectors

When two rays have the same endpoint, an angle is created.

Here, **Always put the vertex (the common endpoint of the two rays) in the middle of the three points.** It doesn’t matter which side point is written first.

An **angle bisector** is a ray 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.

Label equal angles with ** angle markings**, as shown below.

**Investigation:** Constructing an Angle Bisector

- Draw an angle on your paper. Make sure one side is horizontal.
- Place the pointer on the vertex. Draw an arc that intersects both sides.
- Move the pointer to the arc intersection with the horizontal side. Make a second arc mark on the interior of the angle. Repeat on the other side. Make sure they intersect.
- Connect the arc intersections from #3 with the vertex of the angle.

#### Labeling Angles

How many angles are in the picture below? Label each one two different ways.

There are three angles with vertex

So, the three angles can be labeled,

#### Measuring Angles

What is the measure of each angle?

From the picture, we see that the angles are congruent, so the given measures are equal.

To find the measure of

Because

#### Identifying Angle Bisectors

Is

Yes,

### Examples

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

#### Example 1

You should have corresponding markings on

#### Example 2

You should have corresponding markings on

#### Example 3

3. Use algebra to determine the value of d:

The square marking means it is a

### Review

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

- What is the angle bisector of
∠TPR ? - What is
m∠QPR ? - What is
m∠TPS ? - What is
m∠QPV ?

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
90∘ . - An angle bisector divides an angle into three congruent angles.
- Congruent angles have the same measure.

In Exercises 11-15, use the following information:

- Make a sketch.
- Find \begin{align*}m \angle QOP\end{align*}
- Find \begin{align*}m \angle QOT\end{align*}
- Find \begin{align*}m \angle ROQ\end{align*}
- Find \begin{align*}m \angle SOP\end{align*}

### Review (Answers)

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