Chapter 4.3, Problem 4PSA

To determine

To Prove: $XR\perp YZ.$

Explanation of Solution

Given information: XR bisects $\angle YXZ,\angle Y=\angle Z.$

Formula used: Congruency of triangles, properties of triangle, linear pair axiom.

Proof:

  

  $\begin{array}{l}XR\text{bisects}\angle YXZ\text{and}\angle Y=\angle Z\text{(Given)}\\ \therefore \angle YXR=\angle ZXRandXZ=XY\text{(Sides opposite to equal angle are equal)}\\ \therefore \Delta XYR\cong \Delta XZR(\text{ASA})\\ \therefore \angle XRY=\angle XRZ(CPCT)\\ \angle XRY=\angle XRZ={180}^{\circ }\text{(linear pair axiom)}\\ \therefore \angle XRY=\angle XRZ={90}^{\circ }\\ \therefore XR\perp YZ\end{array}$