Given:
In parallelogram PQRS, PO and QO are bisectors of angle P and angle Q respectively.

To Find:
The measure of angle POQ.

Solution:

Let's analyze the given information step by step.

Step 1: Properties of a Parallelogram
- In a parallelogram, opposite sides are equal in length.
- In a parallelogram, opposite angles are equal in measure.
- The sum of interior angles of a parallelogram is 360 degrees.

Step 2: Angle Bisectors
- An angle bisector divides an angle into two equal parts.
- In this case, PO bisects angle P and QO bisects angle Q.

Step 3: Identify Angle Measures
- Let's assume the measure of angle P is x degrees.
- Since PO bisects angle P, the measure of angle POQ will be x/2 degrees.
- Similarly, if we assume the measure of angle Q is y degrees, the measure of angle QOP will be y/2 degrees.

Step 4: Properties of a Parallelogram
- In a parallelogram, opposite angles are equal in measure.
- Therefore, angle QOP and angle SPQ are equal, and angle POQ and angle PSQ are equal.

Step 5: Applying the Properties
- In parallelogram PQRS, angle QOP and angle SPQ are equal, and angle POQ and angle PSQ are equal.
- Since angle QOP and angle SPQ are equal, we can write y/2 = y/2.
- Similarly, since angle POQ and angle PSQ are equal, we can write x/2 = x/2.

Step 6: Solving for Angle POQ
- We have the equation x/2 = x/2.
- By multiplying both sides by 2, we get x = x.
- This means the measure of angle P is equal to the measure of angle Q.
- Therefore, the measure of angle POQ is equal to the measure of angle P, which is x degrees.

Step 7: Conclusion
- The measure of angle POQ is x degrees, where x is the measure of angle P in parallelogram PQRS.

Sum of all angles of a parall. is 360' and sum of two adjacent angles is 180' because interior angles on the same side of two parallel lines is 180' since PO and QO are the bisectors so 1/2angle P +1/2angle Q=1/2 (angle P+Q)=180'/2 =90'=ANGLE QPO+ANGLE PQO IN TRIANGLE POQ ANGLE QPO +POQ +PQO=180' SO ANGLE POQ =180'-90'=90'