Chapter 3.2, Problem 24PSB

To determine

To prove : The congruency of $\Delta VPY\cong \Delta ZPY$ is to be proved

Explanation of Solution

Given information : The following information has been given

  $\overrightarrow{PY}$ bisects $\angle VPZ$

  $\angle VPY={(2x+7)}^0$

  $\angle ZPY={(3x-9)}^0$

  $PZ=\frac{1}{2}x+5$

  $PV=x-3$

Formula used : If two triangles have two congruent sides, and the angle between them is also congruent, then the triangles are also said to be congruent

Proof : If $\overrightarrow{PY}$ bisects $\angle VPZ$ , then

  $2x+7=3x-9$

Hence,

  $x=16$

Now, we use this value in $PZ=\frac{1}{2}x+5$ and $PV=x-3$

Thus, we get $PV=PZ$

Also, they share a common chord PY. Hence, SAS property mentioned above, the congruency $\Delta VPY\cong \Delta ZPY$ can be proved