I am really stuck on a couple of these.

2 column proof Given: FH is congruent to GE and angle HFG is congruent to angle EGF Prove: triangle GEH is congruent with triangle FHE.

To prove that triangle GEH is congruent to triangle FHE, you can use the congruence theorem known as Side-Angle-Side (SAS).

Here is a step-by-step explanation of how to construct a two-column proof for this theorem:

1. Start by writing the given information and labeling the figure. You are given that FH is congruent to GE and angle HFG is congruent to angle EGF.

Given: FH ≅ GE, ∠HFG ≅ ∠EGF

2. Identify the parts of the triangles that are congruent based on the given information. In this case, FH ≅ GE tells us that segments FH and GE are congruent, and ∠HFG ≅ ∠EGF tells us that angles HFG and EGF are congruent.

3. Write the congruence statements corresponding to the congruent parts. We can say:

FH ≅ GE (Given)
∠HFG ≅ ∠EGF (Given)

4. Use the Side-Angle-Side (SAS) congruence theorem to conclude that triangles GEH and FHE are congruent. The SAS theorem states that if two triangles have two pairs of sides that are congruent and the included angles are congruent, then the triangles are congruent.

5. Add the congruence statement for the triangles to your proof. You can write:

Triangle GEH ≅ Triangle FHE (By SAS)

By following these steps and filling out the two-column proof, you will have a formal, logical explanation of why triangle GEH is congruent to triangle FHE.