The bisector of the angles of a parallelogram enclose a ..................

Options

A. Parallelogram
B. square
C. Rhombus
D. Rectangle

Answer

To solve the question "The bisector of the angles of a parallelogram enclose a ..................", we will follow these steps:

### Step 1: Understand the Problem
We need to determine what type of quadrilateral is formed by the angle bisectors of a parallelogram.

### Step 2: Draw a Diagram
Draw a parallelogram ABCD. Label the angles A, B, C, and D. Then, draw the angle bisectors of angles A, B, C, and D, and label the points where these bisectors intersect as P, Q, R, and S.

### Step 3: Analyze the Angles
Since ABCD is a parallelogram, we know that:
- Angle A + Angle D = 180° (consecutive angles)
- Angle B + Angle C = 180° (consecutive angles)

### Step 4: Bisect the Angles
The angle bisectors divide each angle into two equal parts:
- Let the angle bisector of angle A be divided into two angles: \( \frac{1}{2}A \) and \( \frac{1}{2}A \).
- Similarly, for angle D, it will be \( \frac{1}{2}D \) and \( \frac{1}{2}D \).

### Step 5: Sum of Angles at Intersection
At point S (intersection of the bisectors of angles A and D), we have:
\[ \frac{1}{2}A + \frac{1}{2}D = \frac{1}{2}(A + D) = \frac{1}{2}(180°) = 90° \]
Thus, angle DAS + angle ADS = 90°.

### Step 6: Analyze Triangle ASD
In triangle ASD:
\[ \text{Angle ADS} + \text{Angle DAS} + \text{Angle ASD} = 180° \]
Substituting the known values:
\[ 90° + \text{Angle ASD} = 180° \]
This implies:
\[ \text{Angle ASD} = 90° \]

### Step 7: Vertically Opposite Angles
Since angle ASD = 90°, angle RSP (which is vertically opposite to angle ASD) is also 90°.

### Step 8: Find Other Angles
Using the same reasoning, we can find that:
- Angle RQP = 90°
- Angle SRQ = 90°
- Angle QPS = 90°

### Step 9: Conclusion
All angles of quadrilateral PQRS are 90°. Therefore, PQRS is a rectangle.

### Final Answer
The bisector of the angles of a parallelogram encloses a **rectangle**.

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