Question

Given:$▱\mathrm{PQRS};$localid="1637659881844" $\overline{PJ}\cong \overline{RK}$

Prove: localid="1637659923835" $\overline{SJ}\cong \overline{QK}$

Short answer

It is proved that $\overline{SJ}\cong \overline{QK}$.

Step-by-step solution

Step 1. Apply property of parallelogram.

The opposite sides of a parallelogram are congruent.

From the given figure, it can be observed that $\overline{SP}$and $\overline{RQ}$are opposite sides of a $▱PQRS$.

Therefore,$\overline{SP}\cong \overline{RQ}$

Step 2. Apply property of parallelogram.

The opposite angles of a parallelogram are congruent.

From the given figure, it can be observed that $\angle P$and $\angle R$are opposite angles of a $▱PQRS$.

Therefore, $\angle P\cong \angle R$.

Step 3. Apply SAS postulate.

If two sides and the included angle of one triangle are congruent to two sides and included angle of another triangle, then the triangles are congruent.

Here, $\overline{PJ}\cong \overline{RK}$, $\angle P\cong \angle R$and $\overline{SP}\cong \overline{RQ}$, then by SAS postulate $\Delta SPJ\cong \Delta KRQ$.

Step 4. Description of step.

As $\Delta SPJ\cong \Delta KRQ$, then by corresponding parts of congruent triangles, $\overline{SJ}\cong \overline{QK}$.

Hence it is proved that $\overline{SJ}\cong \overline{QK}$.