5.3 Use Angle Bisectors of Triangles

Theorem 5.5: Angle Bisector TheoremIf a point is on the bisector of an angle, then it is equidistant from the two _______ of the angle.sides A

C

B

D

ABDB and BAC bisects AD If _____. DB then ,ACDC and DC

5.3 Use Angle Bisectors of Triangles

Theorem 5.6: Converse of the Angle Bisector TheoremIf a point is in the interior of an angle and is equidistant from the sides of the angle, then it lies on the _________ of the angle.bisector

and ACDC and ABDB If BAC. _______ AD then DC, DB bisects

A

C

B

D

Because EC ____, ED _____, and EC = ED = 21, BE bisects CBD by the ____________________________ _________.

5.3 Use Angle Bisectors of Triangles

Example 1 Use the Angle Bisector TheoremUse the Angle Bisector Theorem

Find the measure of CBE.Find the measure of CBE.

B

D

C

E21

21

31o

BC BDSolution

Converse of the Angle BisectorTheorem

____. _____CBE So, mm DBE o31

5.3 Use Angle Bisectors of Triangles

Example 2 Solve a real-world problemSolve a real-world problem

WebWeb A spider’s position on its web A spider’s position on its web relative to an approaching fly and the relative to an approaching fly and the opposite sides of the web form congruent opposite sides of the web form congruent angles, as shown. Will the spider have to angles, as shown. Will the spider have to move farther to reach a fly toward the move farther to reach a fly toward the right edge or the left edge?right edge or the left edge?

F

L

R

SolutionThe congruent angles tell you that the spider is on the _________ of LFR. bisectorBy the _________________________, the spider is equidistant from FL and FR.

Angle Bisector Theorem

So, the spider must move the ____________ to reach each edge.

same distance

From the Converse of the Angle Bisector Theorem, you know that P lies on the bisector of J if P is equidistant from the sides of J, so then _____ = _____.

5.3 Use Angle Bisectors of Triangles

Example 3 Use algebra to solve a problemUse algebra to solve a problem

Solution

PK

For what value of x does P lie on the bisector of J?

K

J

P

L

x + 1

2x – 5 PL

Set segment Set segment lengths equal.lengths equal.

_____ = _____PK PL______ = _______ Substitute expressions for Substitute expressions for

segment lengths.segment lengths.

1x

52 x

Solve for Solve for xx..______ = _______1 5x

x__6Point P lies on the bisector of J when x = ____. 6

5.3 Use Angle Bisectors of Triangles

Theorem 5.7: Concurrency of Angle Bisector of a TheoremThe angle bisector of a triangle intersect at a point that is equidistant from the sides of the triangle.

A

B

C

P

E

D

Fangle are CP and ,BP ,AP If

thenABC, of bisectors ____. ____ PD PE PF

5.3 Use Angle Bisectors of Triangles

Example 4 Use the concurrency of angle bisectorsUse the concurrency of angle bisectors

In the diagram, L is the incenter of FHJ. Find LK.

I

H

K

G

F

JL

1215

By the Concurrency of Angle Bisectors of a Triangle Theorem, the incenter L is __________ from the sides of FHJ.equidistantSo to find LK, you can find ___ in LHI.LIUse the Pythagorean Theorem. Pythagorean Pythagorean

TheoremTheorem___ = ________2c 22 ba

Substitute Substitute known values.known values.___ = ________

215 22 12LI Simplify.Simplify.___ = ____81 2LISolve.Solve.___ = ____9 LI

Because ____ = LK, LK = _____.LI 9

5.3 Use Angle Bisectors of TrianglesCheckpoint. In Exercise 1 and 2, find the value Checkpoint. In Exercise 1 and 2, find the value xx. .

1.

xo

25o

Because the segments opposite the angles are perpendicular and congruent, by the Converse of the Angle Bisector Theorem, the ray bisects the angle.

So, the angles are congruent, and 25x

5.3 Use Angle Bisectors of TrianglesCheckpoint. In Exercise 1 and 2, find the value Checkpoint. In Exercise 1 and 2, find the value xx. .

By the Angle Bisector Theorem, the two segments are congruent.

2.

7x + 3

8x

xx 837 x3

5.3 Use Angle Bisectors of TrianglesCheckpoint. In Exercise 1 and 2, find the value Checkpoint. In Exercise 1 and 2, find the value xx. .

3. Do you have enough information to conclude that AC bisects DAB?

AC

D

B

No, .90 equal are ADC and ABC that knowmust you o mm

5.3 Use Angle Bisectors of TrianglesCheckpoint. In Exercise 1 and 2, find the value Checkpoint. In Exercise 1 and 2, find the value xx. .

3. In example 4, suppose you are not given HL or HI, but you are given that JL = 25 and JI = 20. Find LK.

I

H

K

G

F

JL

1215

25

20

222 cba 2LI 220 2252LI 400 625

2LI 225LI 15

LK 15

5.3 Use Angle Bisectors of Triangles

Pg. 289, 5.3 #1-12