Chapter 5.1, Problem 10CE

To determine

To state: the theorem or definition which justifies the given statement, “ $\angle ADX\cong \angle CBX$ ”

Answer to Problem 10CE

Definition of parallelogram and Theorem 3-2

Explanation of Solution

Given:

The diagram:

  $ABCD$ is a parallelogram.

Concept Used:

Definition of parallelogram: A parallelogram is a quadrilateral whose both pairs of opposite sides are parallel.

Theorem 3-2: Alternate interior angles are equal.

Consider the given figure:

In this figure it can be observed that $\overline{AB}$ , $\overline{CD}$ and $\overline{AD}$ , $\overline{BC}$ are the pairs of opposite sides of the parallelogram $ABCD$ .

So, be definition of the parallelogram $\overline{AB}\parallel \overline{CD}$ and $\overline{AD}\parallel \overline{BC}$ .

Now, $\overline{AD}\parallel \overline{BC}$ and $\overline{BD}$ is a transversal, so by theorem 3-2,

  $\angle ADB\cong \angle CBD$   (Alternate interior angles)

It can also be written as:

  $\angle ADX\cong \angle CBX$