In the adjoining figure , ` angle AOC = 110 ^(@) , ` Calculate :
` angle ABC `

Answer

To solve the problem, we need to calculate the angle ABC given that angle AOC is 110 degrees. Here’s a step-by-step solution:

### Step-by-Step Solution:

1. **Identify the Given Information:**
- We are given that angle AOC = 110 degrees.

2. **Understand the Geometry:**
- In the figure, O is the center of the circle, and OA, OB, and OC are radii of the circle. Therefore, OA = OB = OC.

3. **Construct the Triangles:**
- Join points A and B to form triangle OAB and triangle OBC.

4. **Use Properties of Isosceles Triangles:**
- Since OA = OB, triangle OAB is isosceles. Thus, the angles opposite to equal sides are equal:
- Let angle OAB = angle OBA = X.

5. **Apply the Triangle Sum Property:**
- In triangle OAB, the sum of angles is 180 degrees:
\[ \text{angle OAB} + \text{angle OBA} + \text{angle AOB} = 180^\circ \]
Substituting the known values:
\[ X + X + \text{angle AOB} = 180^\circ \]
This simplifies to:
\[ 2X + \text{angle AOB} = 180^\circ \quad \text{(Equation 1)} \]

6. **Determine angle AOB:**
- Since OB bisects angle AOC, we have:
\[ \text{angle AOB} = \frac{1}{2} \times \text{angle AOC} = \frac{1}{2} \times 110^\circ = 55^\circ \]

7. **Substitute angle AOB into Equation 1:**
- Now substitute angle AOB into Equation 1:
\[ 2X + 55^\circ = 180^\circ \]
Rearranging gives:
\[ 2X = 180^\circ - 55^\circ = 125^\circ \]
Thus:
\[ X = \frac{125^\circ}{2} = 62.5^\circ \]

8. **Find angle OBC:**
- Since triangle OBC is also isosceles (OB = OC), angle OBC will also be:
\[ \text{angle OBC} = 62.5^\circ \]

9. **Calculate angle ABC:**
- Angle ABC is the sum of angles OBA and OBC:
\[ \text{angle ABC} = \text{angle OBA} + \text{angle OBC} = 62.5^\circ + 62.5^\circ = 125^\circ \]

### Final Answer:
- Therefore, the measurement of angle ABC is **125 degrees**.

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