ABCD is a rhombus in which altitude from point D to the side AB bisects AB. The angles of the rhombus are :

Options

A. `150^(@), 30^(@), 150^(@), 30^(@)`
B. `135^(@), 45^(@), 135^(@), 45^(@)`
C. `116 (1^(@))/(3), 63 (2^(@))/(3), 116 (1^(@))/(3), 62 (2^(@))/(3)`
D. `120^(@), 60^(@), 120^(@), 60^(@)`

Answer

To find the angles of the rhombus ABCD where the altitude from point D to side AB bisects AB, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Properties of a Rhombus**:
- A rhombus is a type of quadrilateral where all four sides are of equal length.
- The diagonals of a rhombus bisect each other at right angles.

2. **Drawing the Diagram**:
- Draw rhombus ABCD.
- Mark points A, B, C, and D.
- Drop a perpendicular from point D to line AB, and let it meet AB at point O (the midpoint of AB).

3. **Identifying Triangles**:
- We can form two right triangles, triangle ADO and triangle BDO.
- Since DO is perpendicular to AB, angle ADO and angle BDO are both 90 degrees.

4. **Using Congruence**:
- Since AO = BO (because O is the midpoint of AB) and AD = DB (as all sides of a rhombus are equal), we can say that triangles ADO and BDO are congruent by the Side-Angle-Side (SAS) criterion.

5. **Finding Angles**:
- Since triangles ADO and BDO are congruent, we have:
- Angle ADO = Angle BDO = 90 degrees
- AD = DB
- This means that both triangles ADO and BDO are isosceles right triangles.

6. **Calculating Angles A and C**:
- In triangle ADO, since it is an isosceles right triangle, the angles A and O are equal.
- Therefore, angle A = angle O = 30 degrees (since the total angle in triangle ADO is 180 degrees, and one angle is 90 degrees).

7. **Calculating Angles B and D**:
- Since angle A + angle B + angle C + angle D = 360 degrees and angles A and C are equal, we can derive:
- Angle A + Angle C = 60 degrees (from the isosceles triangle property).
- Therefore, angle B + angle D = 360 - 60 = 300 degrees.
- Since angle B = angle D, we have 2x = 300 degrees, leading to x = 150 degrees.

8. **Final Angles**:
- Thus, the angles of rhombus ABCD are:
- Angle A = 30 degrees
- Angle B = 150 degrees
- Angle C = 30 degrees
- Angle D = 150 degrees

### Summary of Angles:
- Angle A = 30 degrees
- Angle B = 150 degrees
- Angle C = 30 degrees
- Angle D = 150 degrees