1. Section 5.3
Use Angle Bisectors of Triangles
Remember that an angle bisector is a ______ that divides an angle into two ____________ __________ angles. Remember also that the distance from a point to a line is the length of the ______________ ________ from
the point to the line.
In the diagram,
ray
congruent
adjacent
perpendicular
segment

2. Section 5.3
Use Angle Bisectors of Triangles
Theorem
Theorem 5.5:
Angle Bisector Theorem
If a point is on the angle bisector, then it is equidistant from the sides of the angle.

3. Angle Bisector Theorem
Section 5.3
Use Angle Bisectors of Triangles
Theorem
Theorem 5.6:
Converse of the
Angle Bisector Theorem
If a point is equidistant from the sides of an angle, then it lies on the angle bisector.

4. EXAMPLE 1
Use the Angle Bisector Theorems
Find the measure of GFJ.
FJ bisects GFH, so mGFJ = mHFJ
mGFJ = 42°

5. EXAMPLE 2
Solve a real-world problem
SOCCER
A soccer goalie’s position relative to the ball and goalposts forms congruent angles, as shown. Will the goalie have to move farther to block a shot toward the right goalpost R or the left goalpost L?

6. EXAMPLE 2
Solve a real-world problem
SOLUTION
No. The goalie is standing on the angle bisector and must move the same distance to block either shot.

7. Use algebra to solve a problem
EXAMPLE 3
Use algebra to solve a problem
ALGEBRA
For what value of x does P lie on the bisector of A?
SOLUTION
If P is on the  bisector , then P is equidistant from the sides of A, so BP = CP.
BP = CP
Set segment lengths equal.
x + 3 =
2x –1
Substitute expressions for segment lengths.
4 = x
Solve for x.
Point P is on the bisector of A when x = 4.

8. GUIDED PRACTICE
for Examples 1, 2, and 3
In Exercises 1–3, find the value of x. 1.
x = 15
Two angles are , so the two (blue) segments are  .

9. GUIDED PRACTICE
for Examples 1, 2, and 3
In Exercises 1–3, find the value of x. 2.
Two segments are , so the two s are .
4x – 6 = 3x + 5
x = 11

10. GUIDED PRACTICE
for Examples 1, 2, and 3
In Exercises 1–3, find the value of x. 3.
Two angles are , so the two segments are .
6x – 5 = 5x
x = 5

11. GUIDED PRACTICE
for Examples 1, 2, and 3 4.
Do you have enough information to conclude that
QS bisects PQR? Explain.
No; SR and SP must be  to the sides of the .

12. Angle Bisectors of a Triangle
Section 5.3
Use Angle Bisectors of Triangles
Theorem
Theorem 5.7:
Concurrency of
Angle Bisectors of a Triangle
The angle bisectors of a triangle intersect at a point that is equidistant from the sides.

13. Section 5.3
Use Angle Bisectors of Triangles
The point of concurrency of the three angle bisectors of a triangle is called the ___________. The incenter of a triangle always lies ___________ the triangle.
Because the incenter is equidistant from the ______ of the triangle, a _______ can be drawn inside the triangle. This circle is said to be __________.
incenter
inside
sides
circle
inscribed

14. Use the concurrency of angle bisectors
EXAMPLE 4
Use the concurrency of angle bisectors
In the diagram, N is the incenter of ABC. Find ND.
By Theorem 5.7, Concurrency of Angle Bisectors, the incenter N is equidistant from the sides of ABC. NF = ND. Use the Pythagorean Theorem to find ND.
c = 2
a + b
Pythagorean Theorem
NF + 16 2
20 =
Substitute
400 = NF 2
Multiply
144 = NF 2
Subtract 256
12 = NF
Square root each side
Because NF = ND, ND = 12.

15. GUIDED PRACTICE for Example 4 5. WHAT IF?
In Example 4, suppose you are not given AF or AN, but you are given that BF = 12 and BN = 13. Find ND.
NF + 12 2
13 =
169 = NF 2
25 = NF 2
5 = NF
NF = ND, so ND = 5