Chapter 4.4, Problem 22PSC

To determine

To Prove: AC is perpendicular bisector of BD.

Explanation of Solution

Given information: AB=AD, BC=CD.

Formula used: congruency of triangles, linear pair axiom.

Proof:

  

  $\begin{array}{l}AB=AD,BC=CD(\text{given})\\ AC=AC(\text{Reflective}\text{property)}\\ \therefore \Delta ABC\cong \Delta ADC(\text{SSS})\\ \therefore \angle BAD=\angle DAO(CPCT)\\ OA=OA(\text{Reflective}\text{property)}\\ \therefore \Delta AOB\cong \Delta AOD(\text{SAS})\\ \therefore \angle AOB=\angle AOD,OB=OD(CPCT)\\ \angle AOB+\angle AOD={180}^{\circ }(\text{linear pair axiom})\\ \therefore \angle AOB={90}^{\circ }=\angle AOD\end{array}$