Question

If PQRS is a parallelogram, then which of the following is true?

• A.
  PR = QS
• B.
  PQ = PR
• C.
  $\angle P={50}^{\circ },\angle R={50}^{\circ },\angle Q={130}^{\circ }$
• D.
  $PQ\perp PS$

Answer 1

The correct option is C

$\angle P={50}^{\circ },\angle R={50}^{\circ },\angle Q={130}^{\circ }$

In a parallelogram, the adjacent sides and angles are not equal and no sides are perpendicular to other.

$\angle P+\angle Q={50}^{\circ }+{130}^{\circ }={180}^{\circ }$
Here, $\angle P$ and $\angle Q$ are co-interior angles and their sum is ${180}^{\circ }$.
By the converse of co-interior angles theorem, PS || QR and PQ is the transversal.

$\angle S+\angle P={130}^{\circ }+{50}^{\circ }={180}^{\circ }$
Here, $\angle S$ and $\angle P$ are co-interior angles and their sum is ${180}^{\circ }$.
By the converse of co-interior angles theorem, PQ || RS and PS is the transversal.
Also, the opposite angles of a parallelogram are equal.
Hence, $\angle P=\angle R={50}^{\circ }$.