To complete the proof for the given triangles, let's fill in the missing reasons step by step.

The proof involves showing that $\triangle GHJ \cong \triangle FJI$ using a combination of properties such as corresponding sides and angles of equilateral triangles.

Here's the proof table completed:

Explanation:

• Statement 3: Since $\triangle FGJ$ is equilateral, all sides are congruent, so $FJ \cong GJ$.
• Statement 4: Similarly, in $\triangle HIJ$, all sides are congruent, meaning $HJ \cong IJ$.
• Statement 5: The angles $\angle FJI$ and $\angle GJH$ are corresponding angles in these two equilateral triangles, and since equilateral triangles have equal angles, they are congruent.
• Statement 6: Using the Side-Angle-Side (SAS) congruence postulate, we can conclude that $\triangle GHJ \cong \triangle FJI$.

Would you like further clarification or details on any part of this proof?

Here are some related questions:

1. Why are the angles in equilateral triangles always congruent?
2. What is the SAS congruence postulate?
3. How can you determine if two triangles are congruent using other methods like SSS or ASA?
4. Why is it necessary to prove that corresponding angles are congruent in congruence proofs?
5. Can we use another postulate besides SAS to prove congruence here?

Tip:

In geometry, equilateral triangles are often a key feature in congruence proofs because all sides and angles are congruent, making it easier to apply congruence postulates like SAS or SSS.