PL HELP ME TO SOLVE THIS

Question

Decide if each similarity statement is true or false.

ABF ∼DBC

FBC ∼FDE

ABF∼FDE

FDE∼DBC

Answer 1

Can you see that with the constraints denoted in the picture (several right angles and 2 parallel lines) that there is nothing that constrains the length of segments FE and DE or the sizes of ∠DFE or ∠DEF? I've put a couple of possibilities in the following sketch:

With either of the red line versions of DE and FE, the right angles are still right angles and the parallel lines are still parallel. That means you cannot say triangle FDE is similar to any triangle because it can't be pinned down to a specific triangle. So none of the similarity statements with ΔFDE are necessarily true.

However, ΔFBC has altitude being dropped from its right angle making 3 triangles that are all similar.

ΔBFC ~ ΔDFB ~ ΔDBC. We can also use alternate interior angles, the given right angles, and a common side to use AAS to prove ΔABF ≅ ΔDFB therefore we can say ΔABF ~ ΔDBC