If perpendiculars from any point with an angle on its arms are congruent, prove that it lies on the bisector of that angle.

Explanation

To prove that a point \( P \) lies on the bisector of an angle \( \angle ABC \) when the perpendiculars dropped from \( P \) to the arms of the angle are congruent, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have an angle \( \angle ABC \) with arms \( AB \) and \( AC \). Let \( P \) be a point inside the angle such that the perpendiculars dropped from \( P \) to the arms \( AB \) and \( AC \) are congruent. Denote these perpendiculars as \( PM \) and \( PN \), where \( M \) is the foot of the perpendicular from \( P \) to \( AB \) and \( N \) is the foot of the perpendicular from \( P \) to \( AC \).

2. **Given Information**:
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