Question

For exercises, 14-18 write paragraph proofs.

Given: ABCD is a parallelogram; $\mathbf{DE}\mathbf{=}\mathbf{BF}$.

Prove: AFCE is a parallelogram.

Short answer

As, $\mathbf{CF}\cong \mathbf{AE}$and $\mathbf{AF}\cong \mathbf{CE}$, therefore $\mathbf{AFCE}$is a parallelogram.

Step-by-step solution

Step 1. Observe the given diagram.

The given diagram is:

Step 2. Description of step.

As, $DE=BF$, therefore it can be noticed that:

$\begin{array}{c}DE=BF\\ DE+EF=BF+EF\\ DF=BE\end{array}$

Therefore, $DF\cong BE$.

AsABCD is a parallelogram, therefore $AB\cong CD$,$AB\parallel CD$ and BD is a transversal.

From the given diagram it can be noticed that the angles $\angle FDC$and $\angle EBA$ are the alternate interior angles.

Therefore, $\angle FDC\cong \angle EBA$.

In the triangles $△DCF$and $△BAE$, it can be noticed that $AB\cong CD$, $\angle FDC\cong \angle EBA$and $DF\cong BE$.

Therefore, the triangles $△DCF$and $△BAE$are congruent triangles by using the SAS postulate.

Therefore, by using the corresponding parts of congruent triangles it can be noticed that $CF\cong AE$.

Step 3. Description of step.

AsABCD is a parallelogram, therefore $AD\cong CB$,$AD\parallel BC$ and BD is a transversal.

From the given diagram it can be noticed that the angles $\angle FDA$and $\angle EBC$ are the alternate interior angles.

Therefore, $\angle FDA\cong \angle EBC$.

In the triangles $△DAF$and $△BCE$, it can be noticed that $AD\cong CB$, $\angle FDA\cong \angle EBC$and $DF\cong BE$.

Therefore, the triangles $△DAF$and $△BCE$are congruent triangles by using the SAS postulate.

Therefore, by using the corresponding parts of congruent triangles it can be noticed that $AF\cong CE$.

Step 4. Description of step.

Therefore, it can be noticed that $CF\cong AE$and $AF\cong CE$.

Therefore, it can be said the pairs of opposite sides of the quadrilateral AFCE are congruent.

If the pairs of opposite sides of a quadrilateral are congruent then the quadrilateral is a parallelogram.

As, $CF\cong AE$and $AF\cong CE$, therefore AFCE is a parallelogram.