Chapter 5.2, Problem 14PSA

To determine

To Prove: $\overline{AB}\parallel \overline{CD}$ .

Explanation of Solution

Given:

  

  $\overline{AB}\cong \overline{CD},\overline{BC}\cong \overline{AD}$

Concept Used:

When parallel lines is intersected by a transversal alternate exterior angles are equal.

Proof:

Here, we have

  $\begin{array}{l}\text{In }△ABD\text{ and }△CDB,\\ \overline{AB}\cong \overline{CD}\left(\text{Given}\right)\\ \overline{BC}\cong \overline{AD}\left(\text{Given}\right)\\ \overline{BD}\cong \overline{BD}\left(\text{Common side}\right)\\ \Rightarrow △ABD\cong △CDB\\ \Rightarrow \angle ABD\cong \angle CDB\\ \angle ABD\text{ and }\angle CDB\text{are alternate interior angles}\text{.}\\ \text{Since, we know that when parallel lines is intersected by a transversal alternate interior angles are equal}\text{.}\\ \Rightarrow \overline{AB}\parallel \overline{CD}\text{with transversal }\overline{BD}\end{array}$