If APB and CQD are two parallel lines, then the bisectors of the angles APQ, BPQ, CQP and PQD form Option1: a square Option2: a rhombus Option3: a rectangle Option4: any other parallelogram

Options

A. a square
B. a rhombus
C. a rectangle
D. any other parallelogram

Explanation

To solve the problem, we need to analyze the given information about the angles formed by the two parallel lines APB and CQD, and how their angle bisectors interact.

### Step-by-Step Solution:

1. **Identify the Angles**:
We have two parallel lines, APB and CQD. The angles we are interested in are APQ, BPQ, CQP, and PQD.

2. **Use Properties of Parallel Lines**:
Since APB and CQD are parallel, we know that:
- Angle APQ is equal to angle PQD (alternate interior angles).
- Therefore, we can write:
\[ \angle APQ = \angle PQD \]

3. **Angle Bisectors**:
The angle bisectors of angles APQ and PQD will create two new angles:
- Let \( MPQ \) be the angle bisector of \( APQ \) and \( NQP \) be the angle bisector of \( PQD \).
- Since \( \angle APQ = \angle PQD \), it follows that:
\[ \angle MPQ = \angle NQP \]

4. **Establish Parallel Lines**:
Since \( MPQ \) and \( NQP \) are equal and are both angle bisectors, we can conclude that:
- \( MP \parallel QN \) (as they are corresponding angles).

5. **Consider the Other Angles**:
Similarly, we can analyze angles CQP and BPQ:
- By the same reasoning, we find that the angle bisectors of these angles will also be parallel:
\[ PN \parallel MQ \]

6. **Forming a Parallelogram**:
From the above steps, we have established that:
- \( MP \parallel QN \)
- \( PN \parallel MQ \)
Thus, the figure formed by the bisectors \( MP, QN, PN, \) and \( MQ \) is a parallelogram.

7. **Check for Right Angles**:
To determine if this parallelogram is a rectangle or a square, we need to check if any angle is 90 degrees.
- Since angle CQD is 180 degrees, we can express it as:
\[ \angle CQP + \angle DQP = 180^\circ \]
- Dividing by 2 gives us:
\[ \frac{1}{2} \angle CQP + \frac{1}{2} \angle DQP = 90^\circ \]
- This means that the angles \( MQP \) and \( NQP \) are both 90 degrees.

8. **Conclusion**:
Since we have established that all angles in the quadrilateral formed by the angle bisectors are right angles, we conclude that the shape is a square.

### Final Answer:
The bisectors of angles APQ, BPQ, CQP, and PQD form a **square**.