CK-12 Geometry - Second Edition

5.3. Angle Bisectors in Triangles http://www.ck12.org

In the picture to the right, the blue roads are proposed.

Angle Bisectors

In Chapter 1, you learned that an angle bisector cuts an angle exactly in half. In #1 in the Review Queue above, you
constructed an angle bisector of an 80◦angle. Let’s analyze this figure.

−→

BDis the angle bisector of^6 ABC. Looking at pointD, if we were to drawEDandDF, we would find that they are
equal. Recall from Chapter 3 that the shortest distance from a point to a line is the perpendicular length between
them.EDandDFare the shortest lengths betweenD,which is on the angle bisector, and each side of the angle.

Angle Bisector Theorem:If a point is on the bisector of an angle, then the point is equidistant from the sides of the
angle.

In other words, if

←→

BDbisects^6 ABC,

−→

BE⊥ED, and

−→

BF⊥DF, thenED=DF.

Proof of the Angle Bisector Theorem

Given:

−→

BDbisects^6 ABC,

−→

BA⊥AD, and

−→

BC⊥DC

Prove:AD∼=DC

TABLE5.3:

Statement Reason

1.

−→

BDbisects^6 ABC,

−→

BA⊥AD,

−→

BC⊥DC Given

2.^6 ABD∼=^6 DBC Definition of an angle bisector
3.^6 DABand^6 DCBare right angles Definition of perpendicular lines
4.^6 DAB∼=^6 DCB All right angles are congruent
5.BD∼=BD Reflexive PoC
6. 4 ABD∼= 4 CBD AAS
7.AD∼=DC CPCTC