Chapter 5.6, Problem 15PSB

To determine

To prove:ABCDis a parallelogram.

Explanation of Solution

Given information:

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

Proof:

  

It is given that,

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

Two points determine a line.

Draw $\overline{AC }$ .

By reflexive property, we get

  $\overline{AC }\cong \overline{AC }$

Two triangles are congruent by SSS congruence rule.

  $\Delta ADC\cong \Delta CBA$

Corresponding parts of congruent triangles are congruent.

  $\angle DCA\cong \angle BAC$

Alternate interior angles implies that lines are parallel.

  $\overline{AB }\parallel \overline{DC }$

Corresponding parts of congruent triangles are congruent.

  $\angle DAC\cong \angle BCA$

Alternate interior angles implies that lines are parallel.

  $\overline{AD }\parallel \overline{BC }$

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

ABCD is a parallelogram.