The diagonals Ac and BD of a parallelogram intersects at O. If `angleOAD=40^(@),angleOAB=20^(@)andangleCOD=75^(@)`, then evaluate the following,
`angleABD`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the given information
We have a parallelogram ABCD where the diagonals AC and BD intersect at point O. The angles given are:
- ∠OAD = 40°
- ∠OAB = 20°
- ∠COD = 75°

### Step 2: Identify the relationships between the angles
Since diagonals of a parallelogram bisect each other, we know that:
- ∠OAB and ∠OAD are adjacent angles at point O.
- ∠COD is vertically opposite to ∠AOB.

### Step 3: Find the angle ∠AOB
Using the property of vertically opposite angles:
- ∠AOB = ∠COD = 75°

### Step 4: Use the triangle sum property
In triangle AOB, we know that the sum of the angles is 180°:
- ∠OAB + ∠OAD + ∠AOB = 180°

Substituting the known values:
- 20° + 40° + ∠AOB = 180°
- 60° + ∠AOB = 180°

### Step 5: Solve for ∠AOB
To find ∠AOB:
- ∠AOB = 180° - 60°
- ∠AOB = 120°

### Step 6: Find angle ∠ABD
Now, we need to find angle ∠ABD. Since ∠ABD and ∠AOB are supplementary (they form a linear pair):
- ∠ABD + ∠AOB = 180°
- ∠ABD + 120° = 180°

### Step 7: Solve for ∠ABD
To find ∠ABD:
- ∠ABD = 180° - 120°
- ∠ABD = 60°

### Final Answer
Thus, the angle ∠ABD is 60°.

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