In triangle ABC, `angleABC = 90^@`. BP is drawn perpendicular to AC. If `angleBAP = 50^@`, what is the value (in degree) of `anglePBC`?

Options

A. 30
B. 45
C. 50
D. 60

Answer

To solve the problem, we need to find the value of angle PBC in triangle ABC, where angle ABC is 90 degrees, and BP is drawn perpendicular to AC with angle BAP given as 50 degrees.

### Step-by-Step Solution:

1. **Identify Given Information**:
- Triangle ABC has angle ABC = 90 degrees.
- BP is perpendicular to AC.
- Angle BAP = 50 degrees.

2. **Determine Angle BAC**:
- In triangle ABC, the sum of angles is 180 degrees.
- We know:
\[ \text{Angle ABC} + \text{Angle BAC} + \text{Angle ACB} = 180^\circ \]
- Substituting the known values:
\[ 90^\circ + \text{Angle BAC} + \text{Angle ACB} = 180^\circ \]
- Therefore:
\[ \text{Angle BAC} + \text{Angle ACB} = 90^\circ \]

3. **Find Angle ACB**:
- We know that angle BAP = 50 degrees, which means:
\[ \text{Angle BAC} = \text{Angle BAP} = 50^\circ \]
- Now substituting this into the equation:
\[ 50^\circ + \text{Angle ACB} = 90^\circ \]
- Solving for Angle ACB:
\[ \text{Angle ACB} = 90^\circ - 50^\circ = 40^\circ \]

4. **Analyze Triangle BPC**:
- In triangle BPC, we have:
- Angle BPC = 90 degrees (since BP is perpendicular to AC).
- Angle ACB = 40 degrees (as calculated above).
- The sum of angles in triangle BPC is also 180 degrees:
\[ \text{Angle BPC} + \text{Angle PBC} + \text{Angle PCB} = 180^\circ \]
- Substituting the known values:
\[ 90^\circ + \text{Angle PBC} + 40^\circ = 180^\circ \]

5. **Solve for Angle PBC**:
- Rearranging the equation gives:
\[ \text{Angle PBC} = 180^\circ - 90^\circ - 40^\circ \]
- Thus:
\[ \text{Angle PBC} = 50^\circ \]

### Final Answer:
The value of angle PBC is **50 degrees**.