Question

Given that $▱PQRS$, where $\overline{PX}$bisects $\angle QPR$, $\overline{RY}$bisects $\angle SRP$. Then prove that$RYPX$is a parallelogram.

Short answer

Therefore, RYPX is a parallelogram.

Step-by-step solution

Step 1. Check the figure.

Consider the figure.

Step 2. Step description.

Consider that PQRS is a parallelogram. Thus, the line PS is parallel to the line RQ.

Hence the line PY is parallel to the line RX.

By the alternate interior angle theorem, it is clear that $\angle QPR=\angle SRP$ …... (1)

Step 3. Step description.

Consider the angles $\angle QPR$and $\angle SRP$. Thus, from the figure

$\angle SRP=2\angle YRP$

$\angle QPR=2\angle XPR$

Use the equation (1) as follows:

$\begin{array}{l}2\angle XPR=2\angle YRP\\ \angle XPR=\angle YRP\end{array}$

Now, the angles $\angle XPR$and $\angle YRP$are the alternate interior angles for the line $RY,PX$.

Thus, the lines $RY,PX$are parallel.

Therefore, RYPX is a parallelogram.