In an acute-angled triangle ABC, the altitudes from A, B, C when extended intersect the circumcircle again at points `A_1, B_1, C_1`, respectively. If `angleABC = 45^@` then `angleA_1 B_1 C_1` equals

Options

A. `45^@`
B. `60^@`
C. `90^@`
D. `135^@`

Answer

To find the angle \( \angle A_1 B_1 C_1 \) in triangle \( A_1 B_1 C_1 \) formed by extending the altitudes of triangle \( ABC \) to meet the circumcircle again, we follow these steps:

### Step-by-Step Solution:

1. **Identify Given Information**:
- We have an acute-angled triangle \( ABC \) with \( \angle ABC = 45^\circ \).
- The altitudes from points \( A, B, C \) intersect the circumcircle again at points \( A_1, B_1, C_1 \) respectively.

2. **Draw the Triangle and Circumcircle**:
- Draw triangle \( ABC \) and its circumcircle. Extend the altitudes from points \( A, B, C \) to meet the circumcircle at points \( A_1, B_1, C_1 \).

3. **Analyze Triangle \( A_1 B_1 C_1 \)**:
- We need to find \( \angle A_1 B_1 C_1 \).
- By properties of cyclic quadrilaterals, angles subtended by the same arc are equal.

4. **Use the Angle Properties**:
- Since \( A_1, B_1, C_1 \) lie on the circumcircle, we can use the fact that:
\[ \angle A_1 B_1 C_1 = \angle B C A \]
- Here, \( \angle B C A \) is the angle at vertex \( A \) of triangle \( ABC \).

5. **Calculate \( \angle A_1 B_1 C_1 \)**:
- We know that:
\[ \angle ABC = 45^\circ \]
- Since triangle \( ABC \) is a triangle, we can find the angles at \( A \) and \( C \):
- Let \( \angle A = x \) and \( \angle C = y \).
- By the angle sum property of triangles:
\[ x + 45^\circ + y = 180^\circ \]
\[ x + y = 135^\circ \]
- Therefore, \( \angle A_1 B_1 C_1 = \angle A + \angle C = 135^\circ \).

6. **Conclusion**:
- Thus, the angle \( \angle A_1 B_1 C_1 \) is:
\[ \angle A_1 B_1 C_1 = 135^\circ \]

### Final Answer:
\[ \angle A_1 B_1 C_1 = 135^\circ \]