Chapter 5.6, Problem 14PSB

To determine

To prove:MOPRis a parallelogram.

Explanation of Solution

Given information:

  $MOPR$ is a parallelogram.

  $\overline{MS }\cong \overline{TP }$

Proof:

  

It is given that,

  $MOPR$ is a parallelogram.

  $\overline{MS }\cong \overline{TP }$

Opposite sides of parallelogram are parallel.

  $\overline{RS }\parallel \overline{TO }$

If lines are parallel, then their alternate exterior angles are congruent.

  $\angle RSM\cong \angle PTO$

Opposite sides of parallelogram are congruent.

  $\overline{RS }\cong \overline{TO }$

Two triangles are congruent by SAS congruence rule.

  $\Delta RSM\cong \Delta OTP$

Corresponding parts of congruent triangles are congruent.

  $\begin{array}{l}\overline{RM }\cong \overline{PO }\\ \angle 1\cong \angle 2\end{array}$

Alternate interior angles are congruent implies that lines are parallel.

  $\overline{RM }\parallel \overline{PO }$

If one pairs of opposite sides are both congruent and parallel, then a quadrilateral is parallelogram.