Given the following:

cd←→ is the perpendicular bisector of ab
point g is the midpoint of ab
points e and f lie on cd←→
identy the pair of line segments that must be congruent.

I encountered this question before, it had choices. They were the following:
A) AF ¯ ¯ ¯ ¯ ¯ ≅ BF ¯ ¯ ¯ ¯ ¯ 
B) BG ¯ ¯ ¯ ¯ ¯ ≅ BF ¯ ¯ ¯ ¯ ¯ 
C) AE ¯ ¯ ¯ ¯ ¯ ≅ AF ¯ ¯ ¯ ¯ ¯ 
D) BE ¯ ¯ ¯ ¯ ¯ ≅ BF ¯ ¯ ¯ ¯ ¯ 

Remember that any point on your perpendicular bisector is the same distance from the points on either end of your bisected line segment.  This means that point E should be the same distance to A as it is to B, and point F should also be the same distance from A as it is to B. This means that A is our answer.