Final answer:
To show that line AP bisects the angle between two intersecting lines at A, one can demonstrate that triangles ABP and ACP are congruent, thus proving that AP evenly splits the angle at A.
Step-by-step explanation:
The student's question deals with the geometric property of a point being equidistant from two intersecting lines. We are asked to prove that if point P is equidistant from two lines l and m that intersect at point A, then the line AP must bisect the angle between them. To demonstrate this, consider triangle ABP and triangle ACP where B and C are points on lines l and m respectively, such that AB = AC (since P is equidistant from the lines l and m). With AP being a common side and BP = CP by the definition of P's equidistance, we have two congruent triangles by Side-Side-Side (SSS) Congruence. Therefore, corresponding angles are equal, and angle BAP is equal to angle CAP, meaning line AP bisects the angle at A.