In an isosceles triangle ABC with AB = AC, a line segment XY is drawn parallel to the base BC such that X lies on AB and Y lies on AC. If angle A measures 30 degrees, then what is the measure of angle BXY?

Introduction / Context:
This question tests your understanding of basic angle properties in isosceles triangles and the effect of a line drawn parallel to the base. Such constructions appear frequently in geometry because parallel lines preserve corresponding and alternate interior angles, making it easier to relate interior angles of smaller similar triangles to those of the original triangle.

Given Data / Assumptions:
- Triangle ABC is isosceles with AB = AC.
- Therefore the base is BC and angles at B and C are equal.
- Angle A measures 30 degrees.
- XY is drawn parallel to BC with X on AB and Y on AC.
- We need to find angle BXY at point X.

Concept / Approach:
In an isosceles triangle with AB = AC, the base angles at B and C are equal. The sum of angles in a triangle is 180 degrees, so once we know angle A, we can find the base angles. Drawing XY parallel to BC creates a smaller triangle AXY that is similar to triangle ABC by the angle angle similarity rule because corresponding angles are equal. Therefore, the angle at X corresponding to angle B will be equal to angle B itself.

Step-by-Step Solution:
Step 1: In triangle ABC, sum of angles is 180 degrees, so A + B + C = 180 degrees. Step 2: Given AB = AC, triangle ABC is isosceles with equal base angles, so B = C. Step 3: Angle A is 30 degrees, so B + C = 180 - 30 = 150 degrees. Step 4: Since B = C, each base angle is B = C = 150 / 2 = 75 degrees. Step 5: XY is parallel to BC, so angle BXY is corresponding to angle ABC. Hence angle BXY equals 75 degrees.

Verification / Alternative check:
If you draw a clear diagram, mark angle A as 30 degrees and the base angles as 75 degrees each, and then draw XY parallel to BC, you will see that triangle AXY is a smaller version of ABC. The angle at X between AX and XY is clearly parallel to the angle at B between AB and BC, confirming that angle BXY = angle ABC = 75 degrees.

Why Other Options Are Wrong:
30 degrees is the angle at A, not at X or B, so it cannot equal angle BXY.
60 degrees and 105 degrees do not match any angle directly related to the given isosceles configuration in this setup.
150 degrees is the sum of the two base angles, not a single interior angle formed by the parallel line construction.

Common Pitfalls:
A typical mistake is to confuse the roles of the base angles and the apex angle or to misapply parallel line angle relationships. Some students also try to subtract angles in an incorrect order. Always start by finding all angles in the original triangle and then transfer the appropriate angle measures to the smaller triangle formed by the parallel line.

Final Answer:
The measure of angle BXY is 75 degrees.