1. Use Angle Bisectors of Triangles
5.3
Use Angle Bisectors of Triangles
Theorem 5.5: Angle Bisector Theorem
If a point is on the bisector of an angle, then it is equidistant from the two _______ of the angle. A C B D

2. Use Angle Bisectors of Triangles
5.3
Use Angle Bisectors of Triangles
Theorem 5.6: Converse of the Angle Bisector Theorem
If 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. A C B D

3. Use Angle Bisectors of Triangles
5.3
Use Angle Bisectors of Triangles
Example 1
Use the Angle Bisector Theorem
Find the measure of CBE. B D C E 21
31o
Solution
Because EC ____, ED _____, and EC = ED = 21, BE bisects CBD by the ____________________________ _________.
Converse of the Angle Bisector
Theorem

4. Use Angle Bisectors of Triangles
5.3
Use Angle Bisectors of Triangles
Example 2
Solve a real-world problem F L R
Web A spider’s position on its web relative to an approaching fly and the opposite sides of the web form congruent angles, as shown. Will the spider have to move farther to reach a fly toward the right edge or the left edge?
Solution
The congruent angles tell you that the spider is on the _________ of LFR.
bisector
By the _________________________, the spider is equidistant from FL and FR.
Angle Bisector Theorem
So, the spider must move the ____________ to reach each edge.
same distance

5. Use Angle Bisectors of Triangles
5.3
Use Angle Bisectors of Triangles
Example 3
Use algebra to solve a problem
For what value of x does P lie on the bisector of J?
Solution K J P L
x + 1
2x – 5
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 _____ = _____. PK PL
Set segment lengths equal.
_____ = _____
______ = _______
Substitute expressions for segment lengths.
______ = _______
Solve for x.
Point P lies on the bisector of J when x = ____.

6. Use Angle Bisectors of Triangles
5.3
Use Angle Bisectors of Triangles
Theorem 5.7: Concurrency of Angle Bisector of a Theorem
The angle bisector of a triangle intersect at a point that is equidistant from the sides of the triangle. A B C D E P F

7. Use Angle Bisectors of Triangles
5.3
Use Angle Bisectors of Triangles
Example 4
Use the concurrency of angle bisectors I H K G F J L 12 15
In the diagram, L is the incenter of FHJ. Find LK.
By the Concurrency of Angle Bisectors of a Triangle Theorem, the incenter L is __________ from the sides of FHJ.
equidistant
So to find LK, you can find ___ in LHI. LI
Use the Pythagorean Theorem.
Pythagorean Theorem
___ = ________
Substitute known values.
___ = ________
___ = ____
Simplify.
___ = ____
Solve.
Because ____ = LK, LK = _____. LI

8. Use Angle Bisectors of Triangles
5.3
Use Angle Bisectors of Triangles
Checkpoint. In Exercise 1 and 2, find the value x. xo
25o 1.
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

9. Use Angle Bisectors of Triangles
5.3
Use Angle Bisectors of Triangles
Checkpoint. In Exercise 1 and 2, find the value x. 2.
7x + 3 8x
By the Angle Bisector Theorem, the two segments are congruent.

10. Use Angle Bisectors of Triangles
5.3
Use Angle Bisectors of Triangles
Checkpoint. In Exercise 1 and 2, find the value x.
Do you have enough information to conclude that AC bisects DAB? A C D B
No,

11. Use Angle Bisectors of Triangles
5.3
Use Angle Bisectors of Triangles
Checkpoint. In Exercise 1 and 2, find the value x.
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 J L 15 12 20 25

12. Use Angle Bisectors of Triangles
5.3
Use Angle Bisectors of Triangles
Pg. 289, 5.3 #1-12