Chapter 5.2, Problem 24PSB

To determine

To Prove: $ABCD\text{}\text{is a parallelogram}\text{.}$ .

Explanation of Solution

Given:

  

  $\begin{array}{l}\angle 1\text{supp }\angle 2,\\ \angle 3\text{supp }\angle 2\end{array}$

Concept Used:

When parallel lines is intersected by a transversal interior angles are supplementary.

Proof:

Here, we have

  $\begin{array}{l}\angle 1\text{supp }\angle 2,\\ \Rightarrow \angle 1\text{+}\angle 2={180}^{\circ }\\ \angle 3\text{supp }\angle 2\\ \Rightarrow \angle 3\text{+}\angle 2={180}^{\circ }\\ \text{Since, we know that when parallel lines is intersected by a transversal}\\ \text{interior angles on the same side of transversal are supplementary}\text{.}\\ \Rightarrow \overline{BC}\parallel \overline{AD}\text{with transversal }\overline{AB}\text{and}\\ \Rightarrow \overline{AB}\parallel \overline{DC}\text{with transversal }\overline{AD}\\ \text{If in the quadrilateral pair of opposite sides are parallel,quadrilateral will be }\\ \text{parallelogram}\text{.}\\ \text{Hence, }ABCD\text{}\text{is a parallelogram}\text{.}\end{array}$