Chapter 9, Problem 11CR

To determine

To prove:ABCD is a parallelogram

Explanation of Solution

Given information:The following information has been given

  $EB\cong DF$

  $AG\cong GC$

  $\angle EAG\cong \angle FCG$

Formula used:any quadrilateral with two parallel and equal sides is known as parallelogram

Proof:We know that if $\angle EAG\cong \angle FCG$ , then we can say that

Lines AB and CD are parallel

Thus, we can also say that

  $\angle AEG\cong \angle GFC$

Now, in triangles EAG and CFG, we have

  $\angle AEG\cong \angle GFC$

  $\angle EAG\cong \angle FCG$

  $AG\cong GC$

Thus, we can say that

  $\Delta AEG\cong \Delta GFC$ by ASA property.

Thus, now we can say that

  $AE\cong FC$

Hence, using that $AE\cong FC$ and $EB\cong DF$ , we can say that

  $AB\cong CD$

Also, AB and CD are found to be parallel

Thus, we can now say that the ABCD is parallelogram.