BC and DC are not congruent                assumption
AB is congruent to AD                      given
angle ABC is congruent to angle DAC        given
angle B is congruent to angle D            angles opposite congruent sides are congruent
triangle BAC is congruent to triangle DAC  Angle-Side-Angle
BC is congruent to DC                      corresponding parts of congruent triangles

Assume BC is not congruent to DC.

1. AB ≅ AD        | given
2. ∠BAC ≅ ∠DAC    | given
3. AC ≅ AC        | reflexive property of equality
4.  △BAC ≅ △DAC  | Side-angle-side  (SAS)
5.  BC ≅ DC       | Corresponding parts of congruent triangles. (CPCT)  
6.  BC ≅ DC       | 5 contradicts the assumption that BC is not congruent to DC.

Notice: the theorem was not proved at step 5, because 5 was deduced under the
assumption BC is not congruent to DC.  The theorem was not proved there.  It was
only proved by the statement 6 that the assumption leads to a contradiction.

Edwin