Question

Use the figure below to answer the question that follows:

Intersecting triangles ACE and BDF. They intersect at points G, H, I, and J.

What must be given to prove that ΔBJI ~ ΔCJG?

segment BH is congruent to segment CH and segment BG is congruent to segment CI
∠BIJ ≅ ∠CGJ and ∠JBI ≅ ∠JIB
segment BI is congruent to segment CG and segment JI is congruent to segment JG
∠BIJ ≅ ∠CGJ and ∠BJI ≅ ∠CJG

Answer 1

The answer is D, ∠BIJ ≅ ∠CGJ and ∠BJI ≅ ∠CJG

Answer 2

We have that the initial triangles are equal. Let us check which components remain within the final triangles as the same. The only such components are the angles B and C. We now know that we have to apply a triangle equality criterion. Since all such criteria involve at least one side, we can disregard some answers.
The first answer yields that the triangles BGH and CIH are equal since the have 2 common sides and the respective interior angles are equal. Then we have that BI=GC and also that the respective angles B and C are equal. THen also HIJ=HGJ since they fill to 180 degrees with angles that are equal.
Hence by angle-side-angle the triangles BJI and CJG are equal.
The 3rd answer gives us not enough data since we do not know about the angles BIJ and CGJ