Chapter 5.3, Problem 24PSB

To determine

To Prove: The opposite sides of a parallelogram are congruent.

Explanation of Solution

Given:

  

ABCD as a parallelogram.

Concept Used:

When two parallel lines are intersected by a transversal alternate interior angles are equal

Proof:

Here, we have

  

ABCD as a parallelogram.

Since, when two parallel lines are intersected by a transversal alternate interior angles are equal.

  $\begin{array}{l}\text{In }△BAC\text{ and }△DCA\\ \Rightarrow \angle BAC=\angle DCA\left(\because \text{ Alternate interior angles}\right)\\ \Rightarrow AC=AC\left(\because \text{ Common side}\right)\\ \Rightarrow \angle BCA=\angle DAC\left(\because \text{ Alternate interior angles}\right)\\ \therefore △BAC\cong △DCA\\ \Rightarrow \overline{BC}\cong \overline{AD};\overline{BA}\cong \overline{CD}\end{array}$