Calculate the value of `angleOBC+angleBAC`, if O is the circum-centre of the triangle ABC inscribed in the circle.

Options

A. `60^(@)`
B. `90^(@)`
C. `120^(@)`
D. `130^(@)`

Answer

To solve the problem of calculating the value of \( \angle OBC + \angle BAC \) where \( O \) is the circumcenter of triangle \( ABC \) inscribed in a circle, we can follow these steps:

### Step 1: Understand the Geometry
We have a triangle \( ABC \) inscribed in a circle, with \( O \) being the circumcenter. The circumcenter is the point where the perpendicular bisectors of the sides of the triangle intersect, and it is equidistant from all three vertices of the triangle.

**Hint:** Visualize the triangle and the circumcircle. Identify the circumcenter and the angles involved.

### Step 2: Identify the Angles
We need to find \( \angle OBC \) and \( \angle BAC \).

- \( \angle BAC \) is the angle at vertex \( A \).
- \( \angle OBC \) is the angle formed at point \( O \) when connecting \( O \) to points \( B \) and \( C \).

**Hint:** Label the angles clearly in your diagram for better understanding.

### Step 3: Use the Property of Angles in a Circumscribed Triangle
There is a known theorem regarding the angles in a triangle inscribed in a circle. Specifically, it states that the angle at the circumcenter \( O \) (which is \( \angle OBC \)) and the angle at vertex \( A \) (which is \( \angle BAC \)) are related such that:

\[ \angle OBC + \angle BAC = 90^\circ \]

**Hint:** Recall the theorem about angles related to the circumcenter in a triangle.

### Step 4: Calculate the Sum
From the theorem, we directly have:

\[ \angle OBC + \angle BAC = 90^\circ \]

Thus, the value of \( \angle OBC + \angle BAC \) is \( 90^\circ \).

**Hint:** This is a direct application of the theorem, so no further calculations are needed.

### Conclusion
The final answer is:

\[ \angle OBC + \angle BAC = 90^\circ \]

Therefore, the correct option is **B) 90 degrees**.