A circle of centre O inscribed in a quadrilateral ABCD which touches all the sides of a quadrilateral. If `angleAOB=115^(@)`. Find `angleCOD`.

Answer

To solve the problem, we need to find the angle \( \angle COD \) given that \( \angle AOB = 115^\circ \) in a quadrilateral \( ABCD \) with an inscribed circle.

### Step-by-Step Solution:

1. **Understanding the Geometry**:
- We have a quadrilateral \( ABCD \) with an inscribed circle centered at point \( O \).
- The circle touches all four sides of the quadrilateral.

2. **Identifying Angles**:
- We are given that \( \angle AOB = 115^\circ \).
- By the properties of a cyclic quadrilateral (or a quadrilateral with an inscribed circle), the angles at the center formed by the diagonals are related to the angles at the vertices.

3. **Using the Property of Opposite Angles**:
- The angles at the center \( O \) corresponding to opposite sides of the quadrilateral are supplementary. This means:
\[ \angle AOB + \angle COD = 180^\circ \]

4. **Substituting the Known Value**:
- We know \( \angle AOB = 115^\circ \), so we can substitute this into the equation:
\[ 115^\circ + \angle COD = 180^\circ \]

5. **Solving for \( \angle COD \)**:
- To find \( \angle COD \), we rearrange the equation:
\[ \angle COD = 180^\circ - 115^\circ \]
- Calculating this gives:
\[ \angle COD = 65^\circ \]

### Final Answer:
\[ \angle COD = 65^\circ \]