TxO Math Bowl 2024 - Guts Contest - Set 5 Problem 2

Kayla is learning about triangle congruence, but has not learned about SSA yet. She is given two triangles \triangle ABC and \triangle DEF such that AB = DE = 25, BC = EF = 17, and \angle A = \angle D. When calculating the positive difference between the areas of the two triangles, she gets an answer of 0. However, Kayla's teacher informs her of her mistake, and states, “Triangle ABC and triangle DEF can be different, and in fact, the length of one of AC and DF is exactly 16 greater than the other!” With this information, help Kayla calculate the desired answer.