Question

AB is a line segment and p and q are points on opposite sides of ab such that each of them are equidistantfrom points A andB as shown in the figure show that the line PQ is the perpendicular bisector ofAB

Answer 1

TRIANGLE APQ IS CONGRUENT TO TRIANGLE BPQ 
BY SSS
SO ∠APQ=∠BPQ (CPCT)

TRIANGLE AOP IS CONGRUENT TO TRIANGLE BOP
BY SAS
 O IS POINT OF INTERSECTION OF AB&PQ

SO ∠AOP=∠BOP (CPCT)
 
∠AOP+∠BOP=180
2∠BOP=180
∠BOP=90

HENCE PROOVED