In triangle ABC, angle ABC = 90 degrees and BP is drawn perpendicular to hypotenuse AC. If angle BAP = 50 degrees, what is the measure of angle PBC?

Introduction / Context:
This geometry question explores angle relationships in a right triangle when the altitude is drawn from the right angle to the hypotenuse. It uses triangle similarity and basic angle sum properties. Such configurations appear often in Olympiad and aptitude problems because one right triangle can be split into two smaller similar triangles.

Given Data / Assumptions:
• Triangle ABC is right angled at B, so angle ABC = 90 degrees. • AC is the hypotenuse. • BP is drawn perpendicular to AC, so P is the foot of the altitude from B to AC. • Angle BAP = 50 degrees. • We are asked to find angle PBC.

Concept / Approach:
In a right triangle, the altitude from the right angle to the hypotenuse splits the original triangle into two smaller right triangles, both similar to the original triangle and to each other. Angle BAP is an angle at A between AB and AP, where AP lies along the hypotenuse AC. Since AP is part of AC, the angle between AB and AP equals the angle at A of the original triangle. Therefore angle BAP is the same as angle A. Once we know angle A, we can find angle C, and by considering triangle PBC we can deduce angle PBC.

Step-by-Step Solution:
1. In triangle ABC, angle ABC = 90 degrees. 2. Angle BAP is given as 50 degrees. Since AP lies along AC, angle BAP equals the angle at A of triangle ABC, so angle A = 50 degrees. 3. Using the angle sum of triangle ABC: angle A + angle B + angle C = 180 degrees. 4. Substitute angle A = 50 degrees and angle B = 90 degrees: 50 + 90 + angle C = 180. 5. Simplify: 140 + angle C = 180, so angle C = 40 degrees. 6. Now consider triangle PBC. At point P, BP is perpendicular to AC and C lies on AC, so angle BPC = 90 degrees. 7. In triangle PBC, we already know angle C = angle BCA = 40 degrees, because CP is a segment of AC. 8. Using angle sum in triangle PBC: angle PBC + angle BPC + angle PCB = 180 degrees. 9. Substitute angle BPC = 90 degrees and angle PCB = 40 degrees: angle PBC + 90 + 40 = 180. 10. So angle PBC + 130 = 180, giving angle PBC = 50 degrees.

Verification / Alternative check:
You can also use triangle similarity. Triangles ABC, ABP and PBC are all similar. Angle A in the original triangle equals 50 degrees and corresponds to angle PBC in the smaller triangle at B. This similarity based approach leads directly to angle PBC = 50 degrees without redoing the angle sum in triangle PBC, reinforcing the same result.

Why Other Options Are Wrong:
• 30°: Too small to satisfy the angle sum in triangle PBC with its 90 degree and 40 degree angles. • 45°: Would give a total of 45 + 90 + 40 = 175 degrees, which is less than 180 degrees. • 60°: Would make the sum 60 + 90 + 40 = 190 degrees, exceeding 180 degrees. • 40°: This equals angle C, not angle PBC at B.

Common Pitfalls:
Students often misinterpret angle BAP, thinking it involves some special construction instead of recognising it as the main angle at A. Another mistake is to forget that triangle PBC is right angled at P, not at B, and to use the wrong angle in the sum. Carefully identifying which angles belong to which triangles helps to keep the reasoning consistent.

Final Answer:
The measure of angle PBC is 50°.