Question 543120
I'm not sure that I understand the relationships that you describe among the two triangles. But I think you can prove triangle ABD is congruent to triangle CBD using the following logic:
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Given: Triangles ABD and CBD
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Prove Triangle ABD congruent to Triangle CBD
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Side BA congruent to Side BC ..... Given
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Side AD congruent to side CD ..... Given
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Side BD congruent to side BD ..... Identity
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Triangles ABD & CBD are congruent ... Side/Side/Side congruent to Side/Side/Side
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From your description, I believe that BD is a common side in both triangles and therefore it is a congruent side in both triangles. Two triangles are congruent if all three of their corresponding sides are congruent.  
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If this doesn't work because I have misinterpreted your description of the problem, please re-post your problem and another tutor may be have a better idea of how to help you with this proof.