Given that AB || DC and AD = AB, we can prove that DA ≅ UQ using AA Similarity and the transitive property of equality.
The diagram shows a triangle ABC with side lengths DA, AB, AC, BC, and UQ.
We are given that AB || DC and AD = AB.
Find congruent triangles:
Since AD = AB and angles DAB and CBA are alternate interior angles (formed by a transversal cutting parallel lines), triangle ABD is congruent to triangle CBA by AA Similarity. (SSS would also work here, but AA Similarity is more efficient given the information provided.)
Relate corresponding sides:
Because triangles ABD and CBA are congruent, corresponding sides must be equal. Therefore, BD = AC.
Apply transitivity of equality:
We are given that AD = AB and we just found that BD = AC. Therefore, by the transitive property of equality, AD = AC.
Combine results:
Since AD = AC and UQ is a parallel side to AC, we can conclude that AD ≅ UQ by the converse of the Alternate Interior Angles Theorem.