Chapter 2.7, Problem 7PSA

To determine

To prove : The congruency of $\angle OMR=\angle OPR$ is to be proved

Explanation of Solution

Given information : The following information has been given

  $\angle OMP\cong \angle RPM$

  $\overrightarrow{MP}$ bisects $\angle OMR$

  $\overrightarrow{PM}$ bisects $\angle OPR$

Formula used : Information/data provided in the question that are used to solve this particular category of solution/solution.

Proof : If $\overrightarrow{MP}$ bisects $\angle OMR$ , then we can say that

  $\angle OMP=\angle PMR=a$

And if $\overrightarrow{PM}$ bisects $\angle OPR$ , then we can say that

  $\angle OPM=\angle MPR=b$

Now, if we are given that $\angle OMP\cong \angle RPM$ , then we can say that

  $a=b$

Hence, clubbing all the previous equations together we can say that

  $\angle OPM=\angle MPR=\angle OMP=\angle PMR$

Hence, we can also say that

  $\angle OMP+\angle PMR=\angle OPM+\angle MPR$

Solving and simplifying both sides of equation, we can say that

  $\angle OMR=\angle OPR$

Hence, the given congruency is proved