Final answer:
AC=DC is proven by the Angle Bisector Theorem.
Step-by-step explanation:
In the given scenario, AC bisects ∠BCD, and DC is perpendicular to AD. Additionally, AC is parallel to BD. According to the Angle Bisector Theorem, in a triangle where an angle bisector divides the opposite side, the ratio of the segments of the opposite side is equal to the ratio of the other two sides. In this case, AC bisects ∠BCD, implying that DC/AC = BD/BA. Since AC is parallel to BD, and BA is a transversal, the alternate interior angles are congruent, making BD/BA = DC/AC. Therefore, DC/AC = DC/AC, which implies AC = DC.
Exploring geometric theorems, such as the Angle Bisector Theorem, contributes to a deeper understanding of the relationships between angles and sides in triangles. These theorems are foundational in geometry and help establish logical connections within geometric proofs.