In a parallelogram ABCD, M is the midpoint of BD and BM is bisector of `angleB`. The measure of `angleAMB` is.

Options

A. `45^(@)`
B. `60^(@)`
C. `90^(@)`
D. `120^(@)`

Answer

To solve the problem, we need to find the measure of angle AMB in the given parallelogram ABCD, where M is the midpoint of diagonal BD and BM is the bisector of angle B.

### Step-by-Step Solution:

1. **Draw the Parallelogram**:
- Start by sketching parallelogram ABCD. Label the vertices A, B, C, and D in a clockwise manner.

2. **Identify the Midpoint**:
- Mark point M as the midpoint of diagonal BD. This means that BM = MD.

3. **Angle Bisector**:
- Since BM is the bisector of angle B, we know that angle ABM is equal to angle CBM. Let's denote these angles as angle 1 and angle 2 respectively. Thus, angle ABM = angle CBM.

4. **Identify Alternate Interior Angles**:
- Since AB is parallel to CD (as it is a property of parallelograms), angle CBM (angle 2) is equal to angle DCM (angle 3) because they are alternate interior angles.

5. **Relate Angles**:
- Now we have:
- angle ABM = angle 1
- angle CBM = angle 2
- angle DCM = angle 3
- From the previous steps, we can conclude that:
- angle 1 = angle 3 (since angle 2 = angle 3).

6. **Properties of Parallelogram**:
- In a parallelogram, opposite sides are equal and the diagonals bisect each other. This means that triangles ABM and DCM are congruent (by the Side-Angle-Side postulate).

7. **Conclusion about Shape**:
- Since triangles ABM and DCM are congruent, angle AMB must be equal to 90 degrees because the diagonals of a parallelogram bisect each other at right angles if the parallelogram is a rhombus.

8. **Final Answer**:
- Therefore, the measure of angle AMB is **90 degrees**.