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1.10: Congruent Segments, Midpoints, and Bisectors

Difficulty Level: At Grade Created by: CK-12
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Learning Objectives

  • Understand and apply the Angle Addition Postulate.

Angle, Vertex, and Sides


  • The ________________________ is the shared common endpoint of an angle.
  • In an angle, the two rays are called ______________________.



  • The symbol for naming angles is ___________.

Naming Angles






For the angle with two arcs, you would write JLM.

Reading Check:

Name the following angle three different ways:




Measuring Angles




  • One full rotation is defined as _________________ degrees.
  • The symbol for degrees looks like __________.

Right Angles and Perpendicular Lines




A right angle measures exactly 90.



  • Perpendicular lines intersect at a ___________________ angle.
  • The symbol for perpendicular looks like __________.

Reading Check:

Mark the right angles in the three pictures above.

Construction: Perpendicular Bisector





Reading Check:



Other Types of Angles: Acute, Obtuse, and Straight Angles



  • An ___________________________ angle measures between 0 and 90.
  • An ___________________________ angle measures between 90 and 180.


You can use this information to classify any angle you see.

Reading Check:


______________ _____________ _____________ _____________

Angle Addition Postulate





Example 1

What is mQRT in the diagram below?


mQRS= and mSRT=

So mQRS+mSRT=15+30=45

Therefore, mQRT is 45.

Example 2


To find mLMN, you must subtract mNMO from mLMO:

If mLMO= and mNMO=,

Then mLMOmNMO=8553=32

So \begin{align*}m \angle LMN = 32^\circ\end{align*}.  

Reading Check:

1. Find the measure of angle \begin{align*}\angle WER\end{align*} in the diagram below:

2. Samantha is trying to find the measure of angle \begin{align*}\angle ABD\end{align*} in the figure below. Can you identify Samantha's mistake?

Congruent Angles


  • Congruent angles have the _____________________ measure.


\begin{align*}\angle ABC \cong \angle XYZ\end{align*}

Alternatively, the symbol \begin{align*}m \angle ABC\end{align*} refers to the measure of \begin{align*}\angle ABC\end{align*}, so we could write \begin{align*}m \angle ABC = m \angle XYZ\end{align*} and that has the same meaning as \begin{align*}\angle ABC \cong \angle XYZ\end{align*}.


  • Numbers are equal while objects are _____________________________.


\begin{align*}\angle ABC \cong \angle XYZ\end{align*} (one arc)

\begin{align*}\angle K \cong \angle L\end{align*} (one arc)

\begin{align*}\angle M \cong \angle N\end{align*} (two arcs)

Example 3

The two angles shown below are congruent:

What is the measure of each angle?

Start by setting the angles equal to each other:

\begin{align*}5x+7 &= 3x+23\\ 5x-3x &= 23-7\\ 2x &= 16\\ x &= 8\end{align*}

So, the value of \begin{align*}x\end{align*} is 8. You are not done, however, because the question asks you for the measure of each angle. Use the value of \begin{align*}x\end{align*} to find the measure of one of the angles in the problem (you can choose either angle because we know the angles are congruent):

\begin{align*}m\angle ABC &= 5x+7\\ &= 5(8)+ 7\\ &= 40 + 7\\ &= 47\end{align*}

Since we know \begin{align*}m\angle ABC = m\angle XYZ\end{align*}, both angles measure \begin{align*}47^\circ\end{align*}.

Note: If you wanted to check your work, try substituting the value of \begin{align*}x\end{align*} into the other angle:

\begin{align*}m\angle XYZ &= 3x+23\\ &= 3(\underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;})+ 23\\ &= \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}+ 23\\ &= \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

Did you get the same angle measure?

Construction: Copying A Congruent Angle


Reading Check:



Angle Bisectors


Angle Bisector Postulate

Every angle has exactly one bisector.

Do you remember what a bisector is?

A bisector cuts something in half.


Reading Check:

1. How many angle bisectors can one angle have?










Construction: Angle Bisector


Reading Check:


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