Chapter 4.1, Problem 16PSC

To determine

To Prove: AE bisects BC.

Explanation of Solution

Given information: AB=AC, BD bisects $\angle ABE$ , CD bisect $\angle ACE.$

Formula Used: Congruency of triangles.

Proof:

  $AB=AC$ (Given)

  $\therefore \angle ABE=\angle ACE$ (Angles opposites to equal aides are equal).

  $\therefore 2\angle DBE=2\angle DCE$ (Given BD and CD are angle bisectors)

  $\therefore \angle DBE=\angle DCE.$

  $\therefore BD=DC$ (Sides opposites to equal angles are equal)

  $\begin{array}{l}AD=AD\left(Common\right)\\ \Delta BAD\cong \Delta CAD(SSS)\\ \therefore \angle BAE=\angle CAE(CPCT)\\ \therefore \Delta BAE\cong \Delta CAE(SAS)\\ \therefore BE=CE(CPCT)\end{array}$