Since the triangle is isosceles, this means that AC and BC are congruent (since an isosceles triangle has equal sides). So this is given.
In order to prove that BD = AD, we need to show that the two triangles (that form when the segment CD is drawn) are congruent. Once we've proven that the two triangles are congruent, we can easily show that the corresponding parts are congruent.
Statement | Reason
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1. AC = BC Given
2. CD = CD Reflexive Property of Congruence
3. angle ACD = angle BCD Definition of Angle Bisector
4. triangle ACD = triangle BCD SAS Postulate
5. BD = AD CPCTC
Notes
#1) Remember an "angle bisector" cuts an angle in half. So this means that the two halves of the angle are equal (in this case angles ACD and BCD )
#2) statement 4 uses statements 1, 2, and 3.
#3) CPCTC stands for "Corresponding parts of congruent triangles are congruent".