5.3 Use Angle Bisectors of Triangles

HubarthGeometry

Angle Bisector Theorem

Words If a point is on the bisector of an angle, then it is equidistant from the two sides of the angles.

Symbols If m 1=m 2, then BC BD

12

If then

A . B1

2. B

C

A

D

Find the measure of GFJ.

Ex 1 Use the Angle Bisector Theorem

42HFJm GFJm So, Theorem.Bisector Angle by the GFH bisects

FJ 7,JHJG and JH andFG JG Because

FH

BP = CP Set segment lengths equal.

x + 3 = 2x –1 Substitute expressions for segment lengths.4 = x Solve for x.

Point P lies on the bisector of A when x = 4.

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

Ex 2 Use Algebra to Solve a Problem

Concurrency of Angle Bisector of a Triangle

The angle bisector of a triangle intersects at a point that is equidistant from the sidesof a triangle. B

E

CA

D

FPFPEPDthen

ABC, of bisectors angle are CP and BP ,AP If

In the diagram, N is the incenter of ABC. Find ND.

By the Concurrency of Angle Bisectors of a Triangle Theorem, the incenter N is equidistant from the sides of ABC. So, to find ND, you can find NF in NAF. Use the Pythagorean Theorem.

Ex 3 Use the Concurrency of Angle Bisectors

c = 2 a + b2 2 Pythagorean Theorem

NF + 162220 =2 Substitute known values.

400 = NF + 2562 Multiply.

144 = NF 2

12 = NF

Subtract 256 from each side.

Take the positive square root of each side.

Because NF = ND, ND = 12.

Practice

In Exercises 1–3, find the value of x.

1.

A

B

C

P

15

A

B

C

P2.

11

3.

A

B C

P

5

4. Do you have enough information to conclude that

QS bisects PQR? Explain.

.QPSPand QR SRthat establish toneedyou No,

5. In diagram, suppose you are not given AF or AN, but you are given that BF = 12 and BN = 13. Find ND.

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