Question

For exercises, 14-18 write paragraph proofs.

Given: parallelogram $\mathrm{KGLJ}$; $\mathrm{FK}=\mathrm{HL}$.

Prove: FGHJ is a parallelogram.

Short answer

It is proved that the quadrilateral FGHJ is a parallelogram.

Step-by-step solution

Step 1. Observe the given diagram.

The given diagram is:

Step 2. Description of step.

It is being given that KGLJ is a parallelogram.

In a parallelogram, both diagonals bisect each other.

Therefore, in the parallelogram KGLJ, both diagonals KL and JG intersect each other at the point O.

Therefore, $KO=LO$and $JO=GO$.

Step 3. Description of step.

It is also being given that $FK=HL$.

Therefore, it can be noticed that:

$\begin{array}{c}FK=HL\\ FK+KO=HL+KO\\ FK+KO=HL+LO\left(\because KO=LO\right)\\ FO=HO\end{array}$

Therefore, it can be noticed that $JO=GO$and $FO=HO$.

Therefore, it can be seen that the diagonals FH and JG are bisecting each other at point O.

Therefore, the quadrilateral FGHJ is a parallelogram.