Chapter 5.7, Problem 11PSB

To determine

To prove:That $ACDE$ is an isosceles trapezoid.

Explanation of Solution

Given Information:

  $ABDE$ is a parallelogram

  $BC$ is the base of isosceles $\Delta BCD$

Formula used:

Opposite angles of a parallelogram are equal

Proof:

  $ABDE$ is a parallelogram

We know that, the opposite sides of a parallelogram are equal

  $\Rightarrow AB=DE\text{and}AE=BD\mathrm{...}\left(\text{i}\right)$

  $\Delta BCD$ is an isosceles triangle

  $\Rightarrow BD=CD$

From (i), we get

  $AE=CD$

As $ABDE$ is a parallelogram

  $\Rightarrow ED\parallel AC$

Therefore, in a trapezium $ACDE,$ we get

  $ED\parallel AC\text{and}AE=CD$

Hence, $ACDE$ is an isosceles trapezoid.