Chapter 5.6, Problem 18PSB

To determine

To prove:ABCDis a parallelogram.

Explanation of Solution

Given information:

  $\begin{array}{l}\overline{AF }\parallel \overline{EC },\\ \overline{AF }\cong \overline{EC },\\ \overline{BE }\cong \overline{FD }.\end{array}$

Proof:

  

It is given that,

  $\begin{array}{l}\overline{AF }\parallel \overline{EC },\\ \overline{AF }\cong \overline{EC },\\ \overline{BE }\cong \overline{FD }.\end{array}$

  

The reasons 1 to 6 are as follows:

1.If lines are parallel, then their alternate interior angles are congruent.

  $\angle 1\cong \angle 4$

2. By subtraction property, we get

  $\overline{BF }\cong \overline{ED }$

3. Two triangles are congruent by SAS congruence rule.

  $\Delta AFB\cong \Delta CED$

4. Corresponding parts of congruent triangles are congruent.

  $\angle 2\cong \angle 3$

5. Alternate interior angles are congruent implies that lines are parallel.

  $\overline{AB }\parallel \overline{CD }$

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

ABCD is a parallelogram.