Chapter 5.2, Problem 9CE

To determine

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

Answer to Problem 9CE

It follows that quadrilateral $ABCD$ is a parallelogram.

Explanation of Solution

Given information:

Given figure,

Calculation:

From the figure note that,

  $\begin{array}{l}\angle DAC=\angle BCA\\ \angle CAB=\angle DCA\end{array}$

Note that the angles $\angle DAC$ and $\angle BCA$ 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,

  $AD\parallel BC$

In a similar manner, the angles $\angle CAB$ and $\angle DCA$ are alternate interior angles formed when the transversal $BD$ intersects the pair of line $CD$ and $AB$ .

Since those two angles are congruent, it implies that,

  $AB\parallel CD$

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 parallel, then the quadrilateral is a parallelogram”.