Chapter 5.6, Problem 16PSB

To determine

To prove:ABCDis a parallelogram.

Explanation of Solution

Given information:

  $\begin{array}{l}\overline{AB }\parallel \overline{CD },\\ \overline{AB }\cong \overline{DC }.\end{array}$

Proof:

  

It is given that,

  $\begin{array}{l}\overline{AB }\cong \overline{CD },\\ \overline{AD }\cong \overline{BC }.\end{array}$

If lines are parallel, alternate interior angles are congruent.

  $\begin{array}{l}\angle CDB\cong \angle ABD\\ \angle DCA\cong \angle BAC\end{array}$

Two triangles are congruent by ASA congruence rule.

  $\Delta ABX\cong \Delta CDX$

Corresponding parts of congruent triangles are congruent.

  $\begin{array}{l}\overline{AX}\cong \overline{XC}\\ \overline{BX}\cong \overline{XD}\end{array}$

Vertical angles are congruent.

  $\angle AXD\cong \angle BXC$

Two triangles are congruent by SAS congruence rule.

  $\Delta AXD\cong \Delta CXB$

Corresponding parts of congruent triangles are congruent.

  $\overline{BC}\cong \overline{AD}$

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

ABCD is a parallelogram.