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Congruent Angles and Angle Bisectors

Bisectors split the angle into two equal halves.

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Congruent Angles and Angle Bisectors

Congruent Angles and Angle Bisectors 

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

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

\begin{align*}\overline{BD}\end{align*} is the angle bisector of \begin{align*}\angle ABC\end{align*}

\begin{align*}\angle ABD & \cong \angle DBC\\ m \angle ABD & = \frac{1}{2} m \angle ABC\end{align*}

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 \begin{align*}U\end{align*}. It might be easier to see them all if we separate them out.

So, the three angles can be labeled, \begin{align*}\angle XUY\end{align*} or \begin{align*} \angle YUX, \ \angle YUZ\end{align*} or \begin{align*} \angle ZUY\end{align*}, and \begin{align*}\angle XUZ\end{align*} or \begin{align*} \angle ZUX\end{align*}.

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.

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

To find the measure of \begin{align*}\angle ABC\end{align*}, plug in \begin{align*}x = 8^\circ\end{align*} to \begin{align*}(5x + 7)^\circ\end{align*}.

\begin{align*}& (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.

Identifying Angle Bisectors 

Is \begin{align*}\overline{OP}\end{align*} the angle bisector of \begin{align*} \angle SOT\end{align*}? If \begin{align*}m \angle ROT = 165^\circ\end{align*}, what is \begin{align*}m \angle SOP\end{align*} and \begin{align*}m \angle POT\end{align*}?

Yes, \begin{align*}\overline{OP}\end{align*} is the angle bisector of \begin{align*}\angle SOT\end{align*} according to the markings in the picture. If \begin{align*}m \angle ROT = 165^\circ\end{align*} and \begin{align*}m \angle ROS = 57^\circ\end{align*}, then \begin{align*}m \angle SOT = 165^\circ - 57^\circ = 108^\circ\end{align*}. The \begin{align*}m \angle SOP\end{align*} and \begin{align*}m \angle POT\end{align*} are each half of \begin{align*}108^\circ\end{align*} or \begin{align*}54^\circ\end{align*}.


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

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 #1).

Example 3

3. Use algebra to determine the value of d:

The square marking means it is a \begin{align*}90^\circ\end{align*} angle, so the two angles are congruent. Set up an equation and solve:

\begin{align*}7d-1&=2d +14\\ 5d&=15\\ d&=3\end{align*} 


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

  1. What is the angle bisector of \begin{align*}\angle TPR\end{align*}?
  2. What is \begin{align*}m\angle QPR\end{align*}?
  3. What is \begin{align*}m\angle TPS\end{align*}?
  4. 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.

  1. Every angle has exactly one angle bisector.
  2. Any marking on an angle means that the angle is \begin{align*}90^\circ\end{align*}.
  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: \begin{align*}Q\end{align*} is in the interior of \begin{align*}\angle ROS\end{align*}. \begin{align*}S\end{align*} 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. 

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angle bisector An angle bisector is a ray that splits an angle into two congruent, smaller angles.
Congruent Congruent figures are identical in size, shape and measure.

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