In triangle ABC, angle B is a right angle (90°) and side BC is equal to √3 times side AB. What is the measure of angle A (in degrees)?

Introduction / Context:
This problem involves a right angled triangle and the special ratio of sides that appears in 30°–60°–90° triangles. Questions like this test how well you connect side length ratios with specific angles without having to use trigonometric tables directly.

Given Data / Assumptions:

• Triangle ABC is right angled at B, so ∠B = 90°.
• Side BC is opposite angle A and side AB is opposite angle C.
• The given relation is BC = √3 * AB.
• We need to find the measure of angle A.

Concept / Approach:
In a right angled triangle, trigonometric ratios relate angles to side lengths. For angle A in triangle ABC, tan(A) is defined as (side opposite A) / (side adjacent to A). Here angle A is at vertex A, the side opposite A is BC and one of the sides adjacent to A is AB. So we can use tan(A) = BC / AB. We are given BC / AB = √3. There is a well known special value: tan(60°) = √3. This directly allows us to match the ratio to the angle measure.

Step-by-Step Solution:
Step 1: Identify sides with respect to angle A. Opposite side is BC, adjacent side is AB, and hypotenuse is AC. Step 2: Write the tangent ratio for angle A: tan(A) = BC / AB. Step 3: Substitute the given relation BC = √3 * AB, so BC / AB = √3. Step 4: Therefore tan(A) = √3. Step 5: From standard trigonometric values, tan(60°) = √3. Step 6: Hence A = 60°.

Verification / Alternative Check:
If A = 60° and B = 90°, then C = 180° − 90° − 60° = 30°. In a 30°–60°–90° triangle, the side opposite 60° is √3 times the side opposite 30°. Here BC is opposite A (60°) and AB is opposite C (30°), so BC = √3 * AB, which matches the given condition. This confirms that the result is consistent.

Why Other Options Are Wrong:
45° would give tan(45°) = 1, not √3. An angle of 30° would give tan(30°) = 1/√3, which is the reciprocal of the required value. 90° is impossible because the triangle already has a right angle at B and no triangle can have two right angles.

Common Pitfalls:
Learners sometimes mix up which side is opposite which angle or confuse the ratios for 30° and 60°. Another common mistake is to think that BC is the hypotenuse because it appears in the relation, but the hypotenuse is always opposite the right angle, so it must be AC, not BC. Carefully labeling the triangle before using trigonometric ratios avoids such confusion.

Final Answer:
The measure of angle A is 60°.