Chapter 5.1, Problem 18CE

To determine

Draw two segments that are both parallel and congruent. Connect their endpoints to form a quadrilateral . Find the truth of the quadrilateral.

Answer to Problem 18CE

Opposite sides and opposite angles are congruent and opposite sides are parallel.

Explanation of Solution

Calculation:

Draw the quadrilateral :

  

  $\overline{PQ}\parallel \overline{RS}$

  $\overline{PQ}\cong \overline{RS}$

If we draw a diagonal $\overline{RQ}$ .

Consider the $\Delta PQR\text{ and }\Delta \text{SRQ}$

  $\overline{PQ}\cong \overline{SR}$

  $\angle PQR\cong \angle SRQ$ , since they are alternate interior angles and $\overline{RQ}$ is the diagonal , $\overline{PQ}\parallel \overline{RS}$ .

  $\overline{QR}\parallel \overline{RQ}$

So, by SAS Postulate , $\Delta PQR\cong \Delta \text{SRQ}$ .

Hence, $\overline{PR}\cong \overline{SQ}$ .

  $\angle P\cong \angle S$

Similarly , if we draw diagonal $\overline{PS}$ , $\angle R\cong \angle Q$ .

Also, $\overline{PR}\parallel \overline{QS}$ .

So, we can conclude that opposite sides and opposite angles are congruent and opposite sides are parallel.