Chapter 5.2, Problem 8CE

To determine

To state: whether $ABCD$ must be a parallelogram.

Answer to Problem 8CE

It follows that quadrilateral $ABCD$ is a parallelogram.

Explanation of Solution

Given information:

Given figure,

Calculation:

From the figure note that,

  $\begin{array}{c}AD=BC\\ \angle DCA=\angle BAC\end{array}$

Note that the angles $\angle DCA$ and $\angle BAC$ are alternate interior angles formed when the transversal $AC$ intersects the pair of lines $AD$ and $BC$ .

Since those two angles are congruent, it implies that,

  $AD\parallel BC$

Therefore, it follows that quadrilateral $ABCD$ is a parallelogram.

This follows from the following theorem.

“If one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram”.