Congruent segments theorem

The congruent segments theorem states that if the same segment is added to 2 congruent segments, then the resulting segments will also be congruent.
    
           A ___________________________B
                             C                    D

If AC is congruent to DB, then AD must also be congruent to CB for the reason that you are adding the same segment to congruent segments. This is easily represented by the Addition Property of Equality;

if AC= 2, DB= 2, and CD = 5, then that must mean that AD= AC+CD, which substituted in makes AD= 7
You can also see that for the same reason CB = 7, which means that AD and CB are congruent. No matter what the lengths of the lines are, if AC and DB are congruent, when CD is added, they will still be congruent. Also, if anybody reads this, it took way too long because my cat thinks the keyboard is a pillow.