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Angle Bisectors in Triangles

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Angle Bisectors in Triangles
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What if you were told that \overrightarrow{GJ} is the angle bisector of \angle FGH ? How would you find the length of FJ given the length of HJ ? After completing this Concept, you'll be able to use the Angle Bisector Theorem to solve problems like this one.

Watch This

CK-12 Angle Bisectors

First watch this video.

James Sousa: Introduction to Angle Bisectors

Next watch this video.

James Sousa: Proof of the Angle Bisector Theorem

Then watch this video.

James Sousa: Proof of the Angle Bisector Theorem Converse

Finally, watch this video.

James Sousa: Solving For Unknown Values Using Angle Bisectors

Guidance

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 \overrightarrow{BD} bisects \angle ABC, \ \overrightarrow{BA} \perp \overline{FD} , and, \overrightarrow{BC} \perp \overline{DG} then FD = DG .

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.

Example A

Is Y on the angle bisector of \angle XWZ ?

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

Example B

\overrightarrow{MO} is the angle bisector of \angle LMN . Find the measure of x .

LO = ON by the Angle Bisector Theorem.

4x - 5 &= 23\\4x &= 28\\x &=7

Example C

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

CB=BD by the Angle Bisector Theorem, so we can set up and solve an equation for x .

 x+7 &=2(3x-4)\\x+7 &=6x-8\\15 &=5x\\x &=3

CK-12 Angle Bisectors

Guided Practice

1. \overrightarrow{AB} is the angle bisector of \angle CAD . Solve for the missing variable.

2. Is there enough information to determine if \overrightarrow{A B} is the angle bisector of \angle CAD ? Why or why not?

3. A  108^\circ angle is bisected. What are the measures of the resulting angles?

Answers:

1. CB=BD by the Angle Bisector Theorem, so x=6 .

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

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

Practice

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

Is there enough information to determine if \overrightarrow{A B} is the angle bisector of \angle CAD ? 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 : \overline{AD} \cong \overline{DC} , such that AD and DC are the shortest distances to \overrightarrow{BA} and \overrightarrow{BC}

Prove : \overrightarrow{BD} bisects \angle ABC

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

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