The task is to prove that triangles $\triangle DEF \cong \triangle GHF$ using the given information and the reasoning table. I will help you fill out the missing steps in the table logically.

Given Information:

1. $\angle DEF$ and $\angle GHF$ are right angles.
2. $EF \cong HF$ (side congruence).

Steps to Complete the Proof:

1. Statement 1: $\angle DEF$ is a right angle — Reason: Given.
2. Statement 2: $\angle GHF$ is a right angle — Reason: Given.
3. Statement 3: $\angle DEF \cong \angle GHF$ — Reason: All right angles are congruent.
4. Statement 4: $EF \cong HF$ — Reason: Given.
5. Statement 5: $\triangle DEF \cong \triangle GHF$ — Reason: By the Hypotenuse-Leg (HL) Congruence Theorem.

Final Explanation:

The Hypotenuse-Leg (HL) Congruence Theorem states that two right triangles are congruent if:

1. They have congruent hypotenuses.
2. One pair of legs is congruent.

In this case:

• $EF \cong HF$ (the congruent legs),
• $\angle DEF$ and $\angle GHF$ are right angles.

This is sufficient to conclude $\triangle DEF \cong \triangle GHF$.

Final Completed Table:

Do you want a further explanation of the Hypotenuse-Leg Theorem or any other step?

Related Questions:

1. What is the Hypotenuse-Leg Congruence Theorem?
2. Why are all right angles congruent?
3. Can two triangles be congruent without all three sides being equal?
4. What are other triangle congruence theorems (e.g., SSS, SAS)?
5. How do you prove congruence of triangles using angles and sides?

Tip:

When solving proofs, carefully identify what is given and match them to specific congruence postulates (SSS, SAS, ASA, AAS, or HL).