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calculate ABC and give a reason for your answer

The image shows a problem asking to calculate angle \( \angle ABC \) and provide a reason for the answer. Based on the image, I can see the following details:

- The setup involves a triangle with points \( A \), \( B \), and \( C \).
- There are marked angles \( 30^\circ \) and \( 90^\circ \).
- \( \angle ABC \) is the angle we need to determine.

### Analysis

The triangle depicted is a right-angled triangle, with \( \angle B \) being \( 90^\circ \). In such a triangle:

1. The sum of the angles in any triangle is \( 180^\circ \).
2. Since one of the angles is \( 90^\circ \), the remaining two angles must sum to \( 90^\circ \).

Given:
- \( \angle BAC = 30^\circ \)
- \( \angle ABC = 90^\circ \)

To find \( \angle ACB \):
- Use the fact that \( \angle BAC + \angle ABC + \angle ACB = 180^\circ \).
  
Substituting the known values:

\[ 30^\circ + 90^\circ + \angle ACB = 180^\circ \]

\[ \angle ACB = 180^\circ - 120^\circ = 60^\circ \]

### Conclusion
The calculated \( \angle ABC \) is \( 60^\circ \). The reason is that the angles in a triangle must sum to \( 180^\circ \).

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Would you like more details or have any further questions?

Here are some related questions:
1. How do you determine the sum of angles in any triangle?
2. What properties distinguish right-angled triangles from other triangles?
3. How can the sine, cosine, and tangent ratios help in finding unknown angles?
4. What is the significance of special angles like \( 30^\circ \), \( 45^\circ \), and \( 60^\circ \)?
5. How can you verify the calculated angle using trigonometric functions?

**Tip:** Always check if a triangle is right-angled, as it simplifies calculations with known angle properties.

Learn how to calculate angle ABC in a right-angled triangle with a detailed step-by-step solution. This problem involves applying the angle sum property of a triangle and understanding the properties of right-angled triangles. Suitable for students in Grades 7-9.

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