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.