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.

You need more than a pair of equal sides and a pair of equal angles to prove congruency. Are they right or isosceles triangles?

To prove that triangle GEH is congruent to triangle FHE, we can use the Side-Angle-Side (SAS) congruence criterion.

Given:
- FH is congruent to GE (side)
- angle HFG is congruent to angle EGF (angle)

To prove:
- triangle GEH is congruent to triangle FHE

Proof:
1. FH is congruent to GE (Given)
2. angle HFG is congruent to angle EGF (Given)
3. HG is congruent to EG (corresponding parts of congruent triangles are congruent, from line 1)
4. triangle HFG is congruent to triangle EGF (SAS criterion, using lines 1, 2, and 3)
5. angle GHE is congruent to angle FHE (corresponding parts of congruent triangles are congruent, from line 4)
6. HE is congruent to HE (reflexive property of congruence)
7. triangle GEH is congruent to triangle FHE (SAS criterion, using lines 3, 5, and 6)

Therefore, triangle GEH is congruent to triangle FHE.

Please note that in a step-by-step proof, each statement is justified with a reason or property.

To prove that triangle GEH is congruent to triangle FHE using a 2-column proof, you will need to provide a series of statements and reasons that lead logically from the given information to the desired conclusion. Here's how you can approach it:

Statement:
1. FH is congruent to GE
2. Angle HFG is congruent to angle EGF
3. Segment HE is congruent to itself (Reflexive property)
4. Angle FHE is congruent to itself (Reflexive property)

Reason:
- (Given)
- (Given)
- (Reflexive property)
- (Reflexive property)

Statement:
5. Triangle FHE is congruent to triangle GFE (SAS - Side-Angle-Side congruence)

Reason:
- Side FH is congruent to side GE (Given)
- Angle HFG is congruent to angle EGF (Given)
- Side HE is congruent to itself (Reflexive property)

Statement:
6. Triangle GEH is congruent to triangle FHE (CPCTC - Corresponding Parts of Congruent Triangles are Congruent)

Reason:
- (Statement 5)

After completing the two-column proof with these statements and reasons, you have successfully shown that triangle GEH is congruent to triangle FHE.