AB is a chord to a circle and PAT is the tangent to the circle at A. If `/_BAT = 75^@ and /_BAC = 45^@`, C being a point on the circle, then `/_ABC` is equal to

Options

A. `40^@`
B. `45^@`
C. `60^@`
D. `70^@`

Answer

To solve the problem, we need to find the angle \( \angle ABC \) given the following information:

- \( \angle BAT = 75^\circ \) (the angle between the chord AB and the tangent at point A)
- \( \angle BAC = 45^\circ \) (the angle between the chord AB and the line AC, where C is a point on the circle)

### Step-by-Step Solution:

1. **Identify the Angles**:
- We have \( \angle BAT = 75^\circ \).
- We have \( \angle BAC = 45^\circ \).

2. **Use the Alternate Segment Theorem**:
- According to the Alternate Segment Theorem, the angle formed between a chord and a tangent at the point of contact is equal to the angle in the alternate segment.
- Therefore, \( \angle BCA = \angle BAT = 75^\circ \).

3. **Calculate the Third Angle in Triangle ABC**:
- In triangle ABC, we know two angles:
- \( \angle BAC = 45^\circ \)
- \( \angle BCA = 75^\circ \)
- We can find the third angle \( \angle ABC \) using the fact that the sum of angles in a triangle is \( 180^\circ \):
\[ \angle ABC = 180^\circ - \angle BAC - \angle BCA \]
\[ \angle ABC = 180^\circ - 45^\circ - 75^\circ \]
\[ \angle ABC = 180^\circ - 120^\circ \]
\[ \angle ABC = 60^\circ \]

4. **Conclusion**:
- Therefore, the angle \( \angle ABC \) is \( 60^\circ \).

### Final Answer:
\[ \angle ABC = 60^\circ \]