Chapter 4.7, Problem 28WE

a.

To determine

To prove $AC\cong BE$

Explanation of Solution

Given:

  $AE$ is parallel to $BD$ .

  $BC$ is parallel to $AD$ .

  $AE\cong BC;AD\cong BD$

  

In quadrilateral $ABDE$

  $AE$ is parallel $BD$ (given)

  $\therefore ABDE$ is a trapezoid.

  $\Rightarrow AD=BE$ (diagonals of a regular trapezoid are equal)….(i).

In quadrilateral $ABCD$

  $BC$ is parallel to $AD$ (given)

  $\therefore ABCD$ is a trapezoid.

  $\Rightarrow AC=BD$ (diagonals of a regular trapezoid are equal)….(ii).

From (i) and (ii) ,

  $AD=BD$ and $AC=BE$

  $\therefore AC\cong BE$ .

Hence Proved.

Conclusion:

Hence $AC\cong BE$ .

b.

To determine

To prove that $EC$ is parallel to $AB.$

Explanation of Solution

Given:

  $AE$ is parallel to $BD$ .

  $BC$ is parallel to $AD$ .

  $AE\cong BC;AD\cong BD$

  

In quadrilateral $ABCE$ ,

  $AC=BE$ (proved above)

  $\Rightarrow EC$ is parallel $AB$ .(by the converse of regular trapezoid theorem)

Conclusion:

Hence, $EC$ is parallel to $AB$ .