Chapter 2.6, Problem 16PSC

To determine

To prove : The congruency of angles $\angle XPE\cong \angle ENY$ is to be proven

Explanation of Solution

Given information : The following information has been given

  $\angle 1\cong \angle 2$

  $\overrightarrow{PE}$ bisects $\angle APN$

  $\overrightarrow{NE}$ bisects $\angle ANP$

Formula used : Angle bisector divides any given angle into two equal parts. Thus, if

  $\overrightarrow{PE}$ bisects $\angle APN$ , then we can say that

  $\angle APE\cong \angle EPN$

Proof : We know that $\angle 1\cong \angle 2$ . Thus, their supplementary angles will also be the same

Hence, if $\angle 1\cong \angle 2$ , then we can say that

  $\angle APN\cong \angle ANP$

Now, we use the above mentioned formula for angle bisectors. Hence, if $\angle APN\cong \angle ANP$ , then their bisected angles will also be the same

Thus, we can say that

  $\angle APE\cong \angle PEN\cong \angle ANE\cong \angle ENP$

Hence, we get the two final equations

  $\angle 1\cong \angle 2$

  $\angle APE\cong \angle ANE$

Now, we add up the two equations. Thus,

  $\angle 1+\angle APE\cong \angle 2+\angle ANE$

Simplifying both the sides, we get

  $\angle XPE\cong \angle ENY$

Hence, the required congruency is proved