Page 116 - Math Course 2 (Book 2)

Perpendicular and Angle Bisectors

THEOREM 10.6 Incenter Theorem

The incenter of a triangle is equidistant A
from each side of the triangle. R

Example If K is the incenter of △ABC, then P E
KP = KQ = KR. F K
incenter D

B Q C

Key Concept

Special Segments in Triangles

Name Type Type of Concurrency

perpendicular bisector line, segment, or ray circumcenter

angle bisector line, segment, or ray incenter

median segment centroid

altitude segment orthocenter

Proof:
Let’s Begin Statements Reasons

1. m∠F = 80, m∠E = 30, Given
and DG bisects ∠EDF
Use Angle Bisectors
2. m∠EDF + m∠E + m∠F Angle Sum Theorem
= 180
Example

3. m∠EDF + 30 + 80 = 180 Substitution
Given: m∠F = 80 and m∠E = 30 DG bisects ∠EDF
Prove: m∠DGE = 115 4. m∠EDF = 180–110 = 70 Subtraction Property

F 5. m∠GDE = 35 Definition of angle
bisector
80° G
6. m∠GDE + m∠E + Angle Sum Theorem
m∠DGE = 180

30° 7. 35 + 30 + m∠DGE = 180 Substitution
D E

8. m∠DGE = 180–65 = 115 Subtraction Property

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