In triangle ABC, angle ∠BAC is 90° and the two sides AB and AC are equal in length. What is the measure of angle ∠ABC in degrees?

Introduction / Context:
This question is a straightforward application of angle sum in a triangle and the properties of an isosceles right triangle. It is common in elementary geometry and quantitative aptitude tests.

Given Data / Assumptions:
- Triangle ABC has angle ∠BAC = 90°, so the right angle is at vertex A.
- Sides AB and AC are equal in length, so triangle ABC is isosceles with AB = AC.
- We need to find angle ∠ABC in degrees.

Concept / Approach:
In any triangle, the sum of the three interior angles is 180°. Because AB = AC, the base angles opposite these equal sides, that is, angles at B and C, must also be equal. Thus, ∠ABC = ∠ACB. Knowing that ∠BAC is 90°, the remaining two equal angles must share the remaining 90° equally.

Step-by-Step Solution:
Step 1: Use the angle sum property: ∠BAC + ∠ABC + ∠ACB = 180°. Step 2: Substitute ∠BAC = 90° and let ∠ABC = ∠ACB = x° because AB = AC. Step 3: The equation becomes 90° + x° + x° = 180°. Step 4: Simplify: 90° + 2x° = 180°. Step 5: Subtract 90° from both sides: 2x° = 90°. Step 6: Divide both sides by 2: x° = 45°. Step 7: Therefore ∠ABC = 45°.

Verification / Alternative Check:
Because this is a 45°–45°–90° triangle, the ratio of the sides opposite these angles follows a known pattern. The hypotenuse BC is √2 times any leg. This is consistent with the given right angle at A and equal legs AB and AC. Thus, the computed 45° angles at B and C agree with the standard properties of an isosceles right triangle.

Why Other Options Are Wrong:
Option a, 30°, together with a 90° angle would leave 60° for the third angle, so the base angles would not be equal, contradicting AB = AC.
Option b, 60°, would similarly force the third angle to be 30°, again giving unequal base angles.
Option d, 25°, is incompatible with the triangle angle sum and the requirement that the two non right angles be equal.

Common Pitfalls:
Sometimes students forget which angles are equal in an isosceles triangle and may incorrectly assume the right angle is one of the equal angles. The correct approach is to remember that equal sides lie opposite equal angles. Here AB and AC are equal, so angles at C and B must be equal, not the right angle at A. Using the angle sum property then makes the calculation straightforward.

Final Answer:
The measure of angle ∠ABC is 45°.