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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 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
ABCDEF\begin{align*}\angle ABC \cong \angle DEF\end{align*} Angle ABC\begin{align*}ABC\end{align*} is congruent to angle DEF\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, BD¯¯¯¯¯¯¯¯\begin{align*}\overline{BD}\end{align*} is the angle bisector of ABC\begin{align*}\angle ABC\end{align*}, so ABDDBC\begin{align*}\angle ABD \cong \angle DBC\end{align*} and mABD=12mABC\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

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

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

#### Example 2

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

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

#### Example 3

Write all equal angle statements.

mADBmADF=mBDC=mFDE=45=mADC=90\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.

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

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

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

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

#### Example 5

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

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

### Review

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

1. What is the angle bisector of TPR\begin{align*}\angle TPR\end{align*}?
2. What is mQPR\begin{align*}m\angle QPR\end{align*}?
3. What is mTPS\begin{align*}m\angle TPS\end{align*}?
4. What is mQPV\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 90\begin{align*}90^\circ\end{align*}.
3. An angle bisector divides an angle into three congruent angles.
4. Congruent angles have the same measure.

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

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### Vocabulary Language: English Spanish

TermDefinition
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.