Chapter 4.3, Problem 2PSA

To determine

To Prove: $AD\perp BC$

Explanation of Solution

Given information: AD bisects angle BAC, AB=AC.

Formula used: Congruency of triangles, linear pair axiom.

Proof:

  

  $\begin{array}{l}AD\text{bisects}\angle BAC\text{(Given)}\\ \therefore \angle BAD=\angle CAD.\\ AB=AC\text{(Given)}\\ AD=AD(\text{Reflection property})\\ \therefore \Delta BAD\cong \Delta CAD(\text{SAS})\\ \therefore \angle ADB=\angle ADC(CPCT)\\ \angle ADB+\angle ADC={180}^{\circ }(\text{linear}\text{pair}\text{axiom})\\ \therefore \angle ADB={90}^{\circ }=\angle ADC\\ \therefore AD\perp BC.\end{array}$