Proof:

Given:
- ABCD is a rectangle.
- Diagonal AC bisects angle A as well as angle C.

To prove:
- ABCD is a square.
- Diagonal BD bisects angle B as well as angle D.

Proof:

Let's consider the given rectangle ABCD:

[insert diagram of rectangle ABCD with angles labeled]

1. Diagonal AC bisects angle A:

Since diagonal AC bisects angle A, we have:

[insert diagram of angle A being bisected by diagonal AC]

∠DAC = ∠CAB

2. Diagonal AC bisects angle C:

Similarly, since diagonal AC bisects angle C, we have:

[insert diagram of angle C being bisected by diagonal AC]

∠ACD = ∠ADC

3. Proving ABCD is a square:

To prove that ABCD is a square, we need to show that all four angles of ABCD are right angles.

Since ABCD is a rectangle, we know that opposite sides are parallel and congruent. Therefore, we can conclude that AD is parallel to BC and AB is parallel to DC.

From the given information, we know that diagonal AC bisects angle A and angle C. Therefore, we can deduce that ∠DAC = ∠CAB and ∠ACD = ∠ADC.

Using the fact that opposite angles of a rectangle are congruent, we can now prove that ABCD is a square:

[insert diagram of rectangle ABCD with angles labeled and congruent angles marked]

Since ∠DAC = ∠CAB, and ∠ACD = ∠ADC, we can conclude that ∠DAC + ∠CAB + ∠ACD + ∠ADC = 360°.

Since ABCD is a rectangle, we know that ∠DAC + ∠CAB + ∠ACD + ∠ADC = 360°.

Therefore, ∠DAC + ∠CAB + ∠ACD + ∠ADC = 360°.

Since ∠DAC = ∠CAB and ∠ACD = ∠ADC, we can rewrite the equation as:

∠DAC + ∠DAC + ∠ACD + ∠ACD = 360°.

Simplifying the equation, we get:

2∠DAC + 2∠ACD = 360°.

Dividing both sides by 2, we have:

∠DAC + ∠ACD = 180°.

Since angles DAC and ACD are adjacent angles along the diagonal AC, their sum must be equal to 180°.

This implies that each of these angles is 90°.

Therefore, all four angles of ABCD are right angles, which means ABCD is a square.

4. Diagonal BD bisects angle B as well as angle D:

Since ABCD is a square, the diagonals are congruent and bisect each other at right angles.

Therefore, diagonal BD bisects angle B as well as angle D.

Conclusion:

From the given information, we have proved that ABCD is a square and diagonal BD bisects angle B as well as angle D.