In a circle with centre P, AB is a minor arc. Point Ris a point other than A and B on major arc AB. If `angle ARB = 80^(@),` then ` angle APB=`........

Options

A. `40^(@)`
B. `80^(@)`
C. `160^(@)`
D. `60^(@)`

Answer

To solve the problem, we need to find the angle \( \angle APB \) given that \( \angle ARB = 80^\circ \).

### Step-by-Step Solution:

1. **Understanding the Angles**:
- We have a circle with center \( P \).
- \( AB \) is a minor arc, and \( R \) is a point on the major arc \( AB \).
- The angle \( \angle ARB \) is given as \( 80^\circ \).

2. **Using the Angle Subtended by an Arc**:
- According to the properties of circles, the angle subtended at the center of the circle by an arc is twice the angle subtended at any point on the circumference of the circle.
- This means that \( \angle APB \) (the angle at the center) is related to \( \angle ARB \) (the angle at the circumference).

3. **Applying the Relationship**:
- We can express this relationship mathematically as:
\[ \angle APB = 2 \times \angle ARB \]

4. **Substituting the Known Value**:
- We know that \( \angle ARB = 80^\circ \).
- Therefore, we substitute this value into the equation:
\[ \angle APB = 2 \times 80^\circ = 160^\circ \]

5. **Final Answer**:
- Thus, the value of \( \angle APB \) is \( 160^\circ \).

### Summary of the Solution:
\[ \angle APB = 160^\circ \]