ABCD is a parallelogram. Angle bisector of `angleA` and angle bisector of `angleC` cuts extended side DC and AB at Q and P respectively. If `angleA=50^(@)," find "angleP+angleQ`.

Answer

To solve the problem, we will follow these steps:

1. **Identify the Angles in the Parallelogram**:
In parallelogram ABCD, we know that opposite angles are equal. Given that angle A = 50°, we can conclude that angle C = 50° as well. Since the sum of adjacent angles in a parallelogram is 180°, we can find angles B and D:
- Angle B = 180° - Angle A = 180° - 50° = 130°
- Angle D = 180° - Angle C = 180° - 50° = 130°

2. **Determine the Angles Created by the Angle Bisectors**:
The angle bisector of angle A divides it into two equal parts:
- Angle PAQ = Angle A / 2 = 50° / 2 = 25°.
Similarly, the angle bisector of angle C divides it into two equal parts:
- Angle DCP = Angle C / 2 = 50° / 2 = 25°.

3. **Use the Properties of Parallel Lines**:
Since AB is parallel to DC (as it is a property of parallelograms), the angles formed by the transversal (the angle bisectors) will be equal:
- Angle PAQ = Angle Q = 25° (because they are alternate interior angles).
- Angle DCP = Angle P = 25° (for the same reason).

4. **Calculate the Sum of Angles P and Q**:
Now we can find the sum of angles P and Q:
- Angle P + Angle Q = 25° + 25° = 50°.

Thus, the final answer is:
**Angle P + Angle Q = 50°**.