Chapter 5.5, Problem 14PSA

To determine

To prove: $ACDF$ is not a parallelogram.

Explanation of Solution

Given Information:

  $\begin{array}{l}\angle AFB\cong \angle DCE\\ \Delta AFB\ncong \Delta DCE\end{array}$

Formula used:

Opposite angles of a parallelogram are equal

Opposite sides of a parallelogram are equal

Proof:

Let us assume that $ACDF$ is a parallelogram

As we know that the opposite angles of a parallelogram are equal

  $\Rightarrow \angle CAF=\angle CDF\mathrm{...}\left(\text{i}\right)$

Also, opposite sides of a parallelogram are equal

  $\Rightarrow AF=CD\mathrm{...}\left(\text{ii}\right)$

In $\Delta AFB\text{and}\Delta DCE,$

  $\angle CAF=\angle CDF$ … From (i)

  $\angle AFB=\angle DCE$ … (Given)

  $AF=CD$ … From (ii)

  $\Delta AFB\cong \Delta DCE$ … (By ASA congruence rule)

This is a contradiction to $\Delta AFB\ncong \Delta DCE$

Thus, our assumption is wrong.

Hence, $ACDF$ is not a parallelogram.