Proof:

We are given that ABCD is a rhombus. To prove that diagonal AC bisects angle A and angle C, we need to show that the angles formed by diagonal AC and the sides of the rhombus are congruent.

Angle A:
To prove that diagonal AC bisects angle A, we need to show that angle CAD is congruent to angle DAB.

Proof for angle CAD congruent to angle DAB:
1. Since ABCD is a rhombus, we know that all four sides are congruent.
2. Therefore, AD = AB.
3. Diagonal AC divides the rhombus into two congruent triangles, ADC and ABC (by the Side-Side-Side Congruence theorem).
4. In triangle ADC, AD = AC (as AC is a side of the rhombus).
5. In triangle ABC, AB = AC (as AC is a side of the rhombus).
6. Since AD = AB and AD = AC, we can conclude that AB = AC.
7. In triangle ABD, we have AB = AC and AD = AB, making triangle ABD an isosceles triangle.
8. In an isosceles triangle, the angles opposite the congruent sides are congruent.
9. Therefore, angle CAD is congruent to angle DAB.

Angle C:
To prove that diagonal AC bisects angle C, we need to show that angle BAC is congruent to angle DAC.

Proof for angle BAC congruent to angle DAC:
1. Using the same reasoning as above, we can conclude that triangle BCD is an isosceles triangle.
2. In an isosceles triangle, the angles opposite the congruent sides are congruent.
3. Therefore, angle BAC is congruent to angle DAC.

Diagonal BD:
To prove that diagonal BD bisects angle B and angle D, we can use similar logic as above.

Angle B:
To prove that diagonal BD bisects angle B, we need to show that angle ABD is congruent to angle CBD.

Proof for angle ABD congruent to angle CBD:
1. Using the same reasoning as above, we can conclude that triangle ABD is an isosceles triangle.
2. In an isosceles triangle, the angles opposite the congruent sides are congruent.
3. Therefore, angle ABD is congruent to angle CBD.

Angle D:
To prove that diagonal BD bisects angle D, we need to show that angle BDA is congruent to angle BDC.

Proof for angle BDA congruent to angle BDC:
1. Using the same reasoning as above, we can conclude that triangle BCD is an isosceles triangle.
2. In an isosceles triangle, the angles opposite the congruent sides are congruent.
3. Therefore, angle BDA is congruent to angle BDC.

By proving that diagonal AC bisects angle A and angle C, and diagonal BD bisects angle B and angle D, we have shown that both diagonals of the rhombus bisect the opposite angles.