<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
Skip Navigation
Our Terms of Use (click here to view) have changed. By continuing to use this site, you are agreeing to our new Terms of Use.

Congruent Angles and Angle Bisectors

Bisectors split the angle into two equal halves.

Atoms Practice
Estimated9 minsto complete
Practice Congruent Angles and Angle Bisectors
Estimated9 minsto complete
Practice Now
Turn In
Congruent Angles and Angle Bisectors

Congruent Angles and Angle Bisectors 

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

Here, BA and BC meet to form an angle. An angle is labeled with an “” symbol in front of the three letters used to label it. This angle can be labeled ABC or CBA. 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.

BD¯¯¯¯¯¯¯¯ is the angle bisector of ABC


Label equal angles with angle markings, as shown below.

Investigation: Constructing an Angle Bisector

  1. Draw an angle on your paper. Make sure one side is horizontal.
  2. Place the pointer on the vertex. Draw an arc that intersects both sides.
  3. 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.
  4. 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 U. It might be easier to see them all if we separate them out.

So, the three angles can be labeled, XUY or YUX, YUZ or ZUY, and XUZ or ZUX.

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 ABC, plug in x=8 to (5x+7).


Because mABC=mXYZ, mXYZ=47 too.

Identifying Angle Bisectors 

Is OP¯¯¯¯¯¯¯¯ the angle bisector of SOT? If mROT=165, what is mSOP and mPOT?

Yes, OP¯¯¯¯¯¯¯¯ is the angle bisector of SOT according to the markings in the picture. If mROT=165 and mROS=57, then mSOT=16557=108. The mSOP and mPOT are each half of 108 or 54.








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

Example 1


You should have corresponding markings on A and C.

Example 2


You should have corresponding markings on B and D (that look different from the markings you made in #1).

Example 3

3. Use algebra to determine the value of d:

The square marking means it is a 90 angle, so the two angles are congruent. Set up an equation and solve:



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

  1. What is the angle bisector of TPR?
  2. What is mQPR?
  3. What is mTPS?
  4. What is mQPV?

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.

  1. Every angle has exactly one angle bisector.
  2. Any marking on an angle means that the angle is 90.
  3. An angle bisector divides an angle into three congruent angles.
  4. Congruent angles have the same measure.

In Exercises 11-15, use the following information: Q is in the interior of ROS. S is in the interior of \begin{align*}\angle QOP\end{align*}. \begin{align*}P\end{align*} is in the interior of \begin{align*}\angle SOT\end{align*}. \begin{align*}S\end{align*} is in the interior of \begin{align*}\angle ROT\end{align*} and \begin{align*}m \angle ROT = 160^\circ, \ m \angle SOT = 100^\circ,\end{align*} and \begin{align*}m \angle ROQ = m \angle QOS = m \angle POT\end{align*}.

  1. Make a sketch.
  2. Find \begin{align*}m \angle QOP\end{align*}
  3. Find \begin{align*}m \angle QOT\end{align*}
  4. Find \begin{align*}m \angle ROQ\end{align*}
  5. 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. 

Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes
Please to create your own Highlights / Notes
Show More


angle bisector

An angle bisector is a ray that splits an angle into two congruent, smaller angles.


Congruent figures are identical in size, shape and measure.

Image Attributions

Explore More

Sign in to explore more, including practice questions and solutions for Congruent Angles and Angle Bisectors.
Please wait...
Please wait...