Chapter 5, Problem 24RP

To determine

Prove that $RJMO$ is a parallelogram.

Answer to Problem 24RP

Proved, $RJMO$ is a parallelogram.

Explanation of Solution

Given information:

  

  $P$ is the midpoint $\overline{RO}$ , $K$ is the midpoint of $\overline{JM}$ , $\angle 1\cong \angle 2,\overline{PS}\cong \overline{KS}$

Calculation:

  $\angle 1\cong \angle 2,\overline{PS}\cong \overline{KS}$

  $\begin{array}{l}\overline{RO}\left|\right|\overline{JM}\left[alternate\mathrm{int}eriorangles\cong \right]\\ \angle 3\cong \angle 4\left[\left|\right|lines\to alternate\mathrm{int}eriorangle\cong \right]\\ \angle PSO\cong \angle KSJ\left[verticalanglesarecongruent\right]\\ \Delta PSO\cong \Delta KSJ\left[ASA\right]\\ \overline{PO}\cong \overline{JK}\end{array}$

It is given that $P$ is the midpoint $\overline{RO}$ , $K$ is the midpoint of $\overline{JM}$ ,

  $\overline{RO}\cong \overline{JM}\left[multiplicationproperty\right]$

A quadrilateral with two pairs of opposite congruent sides is a parallelogram,

  $RJMOis\text{}aparalle\log ram$

Hence, proved $RJMO$ is a parallelogram.