# Angle Bisectors in Triangles

## Construction and properties of bisectors, which cut angles in half.

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Angle Bisectors in Triangles

### Angle Bisector Theorem

An angle bisector cuts an angle exactly in half. One important property of angle bisectors is that if a point is on the bisector of an angle, then the point is equidistant from the sides of the angle. This is called the Angle Bisector Theorem.

In other words, if BD\begin{align*}\overrightarrow{BD}\end{align*} bisects ABC, BAFD¯¯¯¯¯¯¯¯\begin{align*}\angle ABC, \ \overrightarrow{BA} \perp \overline{FD}\end{align*}, and, BCDG¯¯¯¯¯¯¯¯\begin{align*}\overrightarrow{BC} \perp \overline{DG}\end{align*} then FD=DG\begin{align*}FD = DG\end{align*}.

The converse of this theorem is also true.

Angle Bisector Theorem Converse: If a point is in the interior of an angle and equidistant from the sides, then it lies on the bisector of that angle.

When we construct angle bisectors for the angles of a triangle, they meet in one point. This point is called the incenter of the triangle.

What if you were told that GJ\begin{align*}\overrightarrow{GJ}\end{align*} is the angle bisector of FGH\begin{align*}\angle FGH\end{align*}? How would you find the length of FJ\begin{align*}FJ\end{align*} given the length of HJ\begin{align*}HJ\end{align*}?

### Examples

#### Example 1

Is there enough information to determine if AB\begin{align*}\overrightarrow{A B}\end{align*} is the angle bisector of CAD\begin{align*}\angle CAD\end{align*}? Why or why not?

No because B\begin{align*}B\end{align*} is not necessarily equidistant from AC¯¯¯¯¯¯¯¯\begin{align*}\overline{AC}\end{align*} and AD¯¯¯¯¯¯¯¯\begin{align*}\overline{AD}\end{align*}. We do not know if the angles in the diagram are right angles.

#### Example 2

A 108\begin{align*} 108^\circ\end{align*} angle is bisected. What are the measures of the resulting angles?

We know that to bisect means to cut in half, so each of the resulting angles will be half of 108\begin{align*}108\end{align*}. The measure of each resulting angle is 54\begin{align*}54^\circ\end{align*}.

#### Example 3

Is Y\begin{align*}Y\end{align*} on the angle bisector of XWZ\begin{align*}\angle XWZ\end{align*}?

If Y\begin{align*}Y\end{align*} is on the angle bisector, then XY=YZ\begin{align*}XY = YZ\end{align*} and both segments need to be perpendicular to the sides of the angle. From the markings we know XY¯¯¯¯¯¯¯¯WX\begin{align*}\overline{XY} \perp \overrightarrow{WX}\end{align*} and ZY¯¯¯¯¯¯¯WZ\begin{align*}\overline{ZY} \perp \overrightarrow{WZ}\end{align*}. Second, XY=YZ=6\begin{align*}XY = YZ = 6\end{align*}. So, yes, Y\begin{align*}Y\end{align*} is on the angle bisector of XWZ\begin{align*}\angle XWZ\end{align*}.

#### Example 4

MO\begin{align*}\overrightarrow{MO}\end{align*} is the angle bisector of LMN\begin{align*}\angle LMN\end{align*}. Find the measure of x\begin{align*}x\end{align*}.

LO=ON\begin{align*}LO = ON\end{align*} by the Angle Bisector Theorem.

4x54xx=23=28=7\begin{align*}4x - 5 &= 23\\ 4x &= 28\\ x &=7\end{align*}

#### Example 5

AB\begin{align*}\overrightarrow{AB}\end{align*} is the angle bisector of CAD\begin{align*}\angle CAD\end{align*}. Solve for the missing variable.

CB=BD\begin{align*}CB=BD\end{align*} by the Angle Bisector Theorem, so we can set up and solve an equation for x\begin{align*}x\end{align*}.

x+7x+715x=2(3x4)=6x8=5x=3\begin{align*} x+7 &=2(3x-4)\\ x+7 &=6x-8\\ 15 &=5x\\ x &=3 \end{align*}

### Review

For questions 1-4, AB\begin{align*}\overrightarrow{AB}\end{align*} is the angle bisector of CAD\begin{align*}\angle CAD\end{align*}. Solve for the missing variable.

Is there enough information to determine if AB\begin{align*}\overrightarrow{A B}\end{align*} is the angle bisector of CAD\begin{align*}\angle CAD\end{align*}? Why or why not?

1. In what type of triangle will all angle bisectors pass through vertices of the triangle?
2. What is another name for the angle bisectors of the vertices of a square?
3. Draw in the angle bisectors of the vertices of a square. How many triangles do you have? What type of triangles are they?
4. Fill in the blanks in the Angle Bisector Theorem Converse.

Given: AD¯¯¯¯¯¯¯¯DC¯¯¯¯¯¯¯¯\begin{align*}\overline{AD} \cong \overline{DC}\end{align*}, such that AD\begin{align*}AD\end{align*} and DC\begin{align*}DC\end{align*} are the shortest distances to BA\begin{align*}\overrightarrow{BA}\end{align*} and BC\begin{align*}\overrightarrow{BC}\end{align*}

Prove: BD\begin{align*}\overrightarrow{BD}\end{align*} bisects ABC\begin{align*}\angle ABC\end{align*}

Statement Reason
1. 1.
2. 2. The shortest distance from a point to a line is perpendicular.
3. DAB\begin{align*}\angle DAB\end{align*} and DCB\begin{align*}\angle DCB\end{align*} are right angles 3.
4. DABDCB\begin{align*}\angle DAB \cong \angle DCB\end{align*} 4.
5. BD¯¯¯¯¯¯¯¯BD¯¯¯¯¯¯¯¯\begin{align*}\overline{BD} \cong \overline{BD}\end{align*} 5.
6. \begin{align*}\triangle ABD \cong \triangle CBD\end{align*} 6.
7. 7. CPCTC
8. \begin{align*}\overrightarrow{B D}\end{align*} bisects \begin{align*}\angle ABC\end{align*} 8.

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### Vocabulary Language: English

TermDefinition
angle bisector An angle bisector is a ray that splits an angle into two congruent, smaller angles.
Angle Bisector Theorem The angle bisector theorem states that if a point is on the bisector of an angle, then the point is equidistant from the sides of the angle.
Angle Bisector Theorem Converse The angle bisector theorem converse states that if a point is in the interior of an angle and equidistant from the sides, then it lies on the bisector of that angle.
incenter The incenter is the point of intersection of the angle bisectors in a triangle.