Chapter 12.7, Problem 13E

To determine

To write: The proof to verify that the construction is valid.

Answer to Problem 13E

The proof to verify the construction is valid is stated below.

Explanation of Solution

Consider $\Delta APQ$ and $\Delta BPQ$ .

  $\begin{array}{c}\overline{AP}=\overline{BP}\\ \overline{AQ}=\overline{BQ}\\ \overline{PQ}=\overline{PQ}\end{array}$

Thus, by SSS congruence theorem.

  $\Delta APQ\cong \Delta BPQ$

Consider $\Delta APM$ and $\Delta BPM$ .

  $\begin{array}{c}\overline{AM}=\overline{BM}\\ \angle APM=\angle BPM\\ \overline{PM}=\overline{PM}\end{array}$

Thus, by SAS congruence theorem.

  $\Delta APM\cong \Delta BMP$

So, by CPCT,

  $\angle AMP\cong \angle BMP$

Also they form a linear pair.

So,

  $\overline{MP}\perp \overline{AB}$

Thus,

  $\angle AMP$ and $\angle BMP$ are right angles.

Hence, proved.