If in a rectangle ABCD, diagonal AC bisect `angleA` as well as `angleC` then ABCD is a

Options

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

Answer

To solve the problem, we need to prove that if in rectangle ABCD, diagonal AC bisects both angle A and angle C, then ABCD is a square.

### Step-by-Step Solution:

1. **Understanding the Properties of a Rectangle**:
- A rectangle has opposite sides that are equal and all angles are 90 degrees.

2. **Given Information**:
- Diagonal AC bisects angle A and angle C.
- This means that angle A is divided into two equal angles, and angle C is also divided into two equal angles.

3. **Setting Up the Angles**:
- Let angle A = 2α (since it is bisected, each part will be α).
- Similarly, let angle C = 2β (since it is bisected, each part will be β).

4. **Using the Properties of a Rectangle**:
- In a rectangle, angle A and angle C are both 90 degrees.
- Therefore, we can set up the equations:
- 2α = 90 degrees → α = 45 degrees
- 2β = 90 degrees → β = 45 degrees

5. **Analyzing the Triangle Formed by the Diagonal**:
- Consider triangle ABC formed by points A, B, and C.
- Since angle A = 45 degrees and angle B = 90 degrees, angle C must also be 45 degrees (because the sum of angles in a triangle is 180 degrees).

6. **Using the Isosceles Triangle Property**:
- In triangle ABC, since angle A = angle C = 45 degrees, it follows that sides AB and BC must be equal (because the angles opposite to equal sides are equal).

7. **Conclusion About the Sides**:
- Since AB = BC and also AD = DC (as opposite sides of a rectangle are equal), we can conclude that all sides of rectangle ABCD are equal (AB = BC = CD = DA).

8. **Final Conclusion**:
- Since all sides are equal and all angles are 90 degrees, rectangle ABCD is a square.

### Final Answer:
ABCD is a square.

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