Chapter 5.1, Problem 16WE

To determine

To prove: $\angle B\cong \angle E$

Explanation of Solution

Given:

The figure:

  $ABCX$ is a parallelogram.

  $DXFE$ is a parallelogram.

Concept Used:

Theorem 5-2: Opposite angles of a parallelogram are congruent.

Theorem 2-3: Vertical angles are congruent.

Consider the given figure:

It is given that $ABCX$ is a parallelogram.

So by theorem 5-2, opposite angle of parallelogram $ABCX$ are congruent. So,

$\angle B\cong \angle AXC$ .... (1)

Also, from the given figure it is clear that $\angle AXC$ and $\angle DXF$ are vertical angles, so by theorem 2-3,

$\angle AXC\cong \angle DXF$ .... (2)

It is also given that $DXFE$ is a parallelogram.

So by theorem 5-2, opposite angle of parallelogram $DXFE$ are congruent. So,

  $\angle DXF\cong \angle E$ .... (3)

Now, from the equation (1), (2) and (3),

  $\begin{array}{c}\angle B\cong \angle AXC\\ \cong \angle DXF\\ \cong \angle E\end{array}$

Hence proved that $\angle B\cong \angle E$