If `APB` and `CQD` are two parallel lines, then the bisectors of the angles `APQ, BPQ, CQP` and `PQD` form

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 parallel lines and the angle bisectors. Here’s a step-by-step solution:

### Step 1: Understand the Configuration
We have two parallel lines, `APB` and `CQD`. The angles formed by these lines with a transversal line `PQ` create angles `APQ`, `BPQ`, `CQP`, and `PQD`.

**Hint:** Visualize the parallel lines and the transversal to understand the angles formed.

### Step 2: Identify the Angle Bisectors
Let’s denote the angle bisectors of the angles:
- The bisector of angle `APQ` as `MP`
- The bisector of angle `BPQ` as `NP`
- The bisector of angle `CQP` as `MQ`
- The bisector of angle `PQD` as `NQ`

**Hint:** Remember that the angle bisector divides the angle into two equal parts.

### Step 3: Establish Relationships Between Angles
Since `MP` is the bisector of angle `APQ`, we have:
- Angle `1` = Angle `5` (where angle `1` is part of angle `APQ` and angle `5` is formed by the bisector)

Similarly, we can establish:
- Angle `2` = Angle `6`
- Angle `3` = Angle `7`
- Angle `4` = Angle `8`

**Hint:** Use the property of angle bisectors to write down equalities for the angles.

### Step 4: Use the Properties of Parallel Lines
Since `APB` and `CQD` are parallel, we can use the Alternate Interior Angles Theorem:
- Angle `APQ` (angle `1` + angle `5`) = Angle `DQK` (angle `4` + angle `8`)

This means:
- Angle `1` + Angle `5` = Angle `4` + Angle `8`

Using the relationships established in Step 3, we can say:
- Angle `1` = Angle `4` and Angle `2` = Angle `3`

**Hint:** Recall that alternate angles are equal when two lines are parallel.

### Step 5: Establish Parallelism of Bisectors
From the equal angles, we can conclude:
- Since angle `1` = angle `4`, the lines `MP` and `NQ` are parallel.
- Since angle `2` = angle `3`, the lines `MP` and `MQ` are also parallel.

**Hint:** Use the properties of angles to determine the relationships between the lines.

### Step 6: Determine the Shape Formed
Since we have established that:
- `MP` is parallel to `NQ`
- `MQ` is parallel to `PN`

This indicates that the figure formed by the bisectors is a parallelogram.

### Step 7: Check for Right Angles
To determine if the parallelogram is a rectangle, we need to check if any angles are right angles. Since the angles formed by the transversal with the parallel lines sum up to 180 degrees, we can conclude that:
- Angle `3` + Angle `4` = 90 degrees.

Thus, the angles formed by the bisectors are right angles, confirming that the parallelogram is a rectangle.

**Hint:** Remember that a rectangle is a special type of parallelogram where all angles are right angles.

### Conclusion
Therefore, the bisectors of the angles `APQ`, `BPQ`, `CQP`, and `PQD` form a rectangle.

**Final Answer:** The bisectors form a rectangle.