The diagonals of rectangle ABCD intersect at O . If `angleAOD=40^(@)` , then find `angleOAD`

Answer

To solve the problem, we need to find the angle \( \angle OAD \) in rectangle \( ABCD \) where the diagonals intersect at point \( O \) and \( \angle AOD = 40^\circ \).

### Step-by-Step Solution:

1. **Understanding the Properties of a Rectangle**:
- In a rectangle, the diagonals bisect each other and are equal in length. Therefore, \( OA = OC \) and \( OB = OD \).

2. **Identifying Angles**:
- We know that \( \angle AOD = 40^\circ \). Since \( O \) is the intersection point of the diagonals, we can also state that \( \angle BOC \) is equal to \( \angle AOD \) because diagonals of a rectangle bisect each other.

3. **Using the Triangle Angle Sum Property**:
- In triangle \( OAD \), the sum of the angles is \( 180^\circ \). Therefore, we can write:
\[ \angle OAD + \angle AOD + \angle ODA = 180^\circ \]

4. **Identifying Angles in Triangle OAD**:
- Since \( OA = OD \) (as mentioned earlier), triangle \( OAD \) is isosceles. This means that the angles opposite to the equal sides are equal:
\[ \angle OAD = \angle ODA \]
- Let's denote \( \angle OAD = \angle ODA = x \).

5. **Setting Up the Equation**:
- Now we can substitute into the triangle angle sum equation:
\[ x + 40^\circ + x = 180^\circ \]
- This simplifies to:
\[ 2x + 40^\circ = 180^\circ \]

6. **Solving for x**:
- Subtract \( 40^\circ \) from both sides:
\[ 2x = 180^\circ - 40^\circ = 140^\circ \]
- Now divide by 2:
\[ x = \frac{140^\circ}{2} = 70^\circ \]

7. **Conclusion**:
- Therefore, \( \angle OAD = 70^\circ \).

### Final Answer:
\[ \angle OAD = 70^\circ \]