In triangle ABC, angle ∠ABC = 90°. From vertex B, a perpendicular BP is dropped to hypotenuse AC. If ∠BAP = 30°, what is the measure (in degrees) of angle ∠PBC?

Introduction / Context:
This problem involves a right triangle with an altitude drawn from the right angle to the hypotenuse. Such configurations generate several smaller right triangles that are similar to the original triangle. The question requires angle chasing rather than numerical calculation. By using the properties of angle bisectors created by the altitude and the fact that the full angle at B is 90 degrees, we can find the value of angle ∠PBC once ∠BAP is known.

Given Data / Assumptions:
- Triangle ABC is right angled at B, so ∠ABC = 90°. - BP is drawn perpendicular to hypotenuse AC, so BP is an altitude from B to AC. - ∠BAP = 30°, where P lies on AC between A and C. - We need to find angle ∠PBC at vertex B. - All angles are interior angles of the relevant triangles.

Concept / Approach:
When a perpendicular is drawn from the right angle vertex to the hypotenuse in a right triangle, the resulting smaller triangles are similar to the original. However, a more straightforward approach here is pure angle chasing. We use the fact that lines AB and CB are perpendicular, so any acute angle between BP and AB determines the complementary angle between BP and BC. The angle ∠BAP is connected to the geometry of the triangle and determines where P lies on AC. A suitable coordinate or similarity argument shows that ∠PBC ends up equal to ∠BAP for this configuration.

Step-by-Step Solution:
Step 1: Note that ∠ABC = 90°, so AB and CB are perpendicular. Step 2: BP is perpendicular to AC, so triangle BAP and triangle BCP are both right angled at P. Step 3: In triangle ABC, let us consider a coordinate model where B is at the origin, AB lies along the horizontal axis and BC along the vertical axis. Step 4: In such a model, A and C lie on the axes and P is the foot of the perpendicular from B to the line AC. Step 5: If we impose the condition that ∠BAP = 30°, the geometry forces a specific ratio between the legs BA and BC that fixes the exact position of P. Step 6: A direct computation or use of similar triangles shows that the angle between BP and BC, which is ∠PBC, equals 30°. Step 7: Thus ∠PBC has the same measure as ∠BAP and evaluates to 30 degrees.

Verification / Alternative check:
Using coordinates we can place B at (0, 0), A at (1, 0) and C at (0, 1 / √3) so that ∠BAP becomes 30 degrees after solving the angle condition. With this configuration, we find the coordinates of P as the perpendicular projection of B onto AC. Computing the angle between BP and BC using dot products shows that cos(∠PBC) equals √3 / 2, which corresponds to 30 degrees. This independent check confirms that ∠PBC is indeed 30°, matching the reasoning from angle chasing.

Why Other Options Are Wrong:
Option 36° does not correspond to any standard complementary or special triangle angle in this configuration. Option 45° would require an isosceles right triangle structure that does not match the given 30° condition at A. Option 60° would make angles at B and at A inconsistent with the requirement that the sum of angles in triangle ABC is 180°.

Common Pitfalls:
One common mistake is to assume that the altitude bisects the hypotenuse or bisects an angle without proof. Another issue is mixing up which angles are complementary. Some learners also try to treat ∠BAP directly as an angle in a 30 60 90 triangle without carefully relating it to ∠PBC. Drawing a clear diagram, marking right angles and using a coordinate or similarity approach helps avoid these errors.

Final Answer:
The required angle ∠PBC is 30°.