https://file.aiquickdraw.com/mathbot/file/17273645371523u0kb2pg.jpg

The image shows a proof where we need to complete the final step. The goal is to prove that:

\[
\frac{GH}{FG} = \frac{IJ}{FJ}
\]

The proof already provides a set of statements and reasons that lead up to this conclusion. Let’s analyze what has been done and finalize the proof:

### Current proof statements:

1. \( HI \parallel GJ \)  – Given  
2. \( \angle F \cong \angle F \)  – Reflexive Property of Congruence  
3. \( \angle FGJ \cong \angle H \)  – Corresponding Angles Theorem  
4. \( \triangle FGJ \sim \triangle FHI \)  – Angle-Angle (AA) Similarity  
5. \( \frac{FH}{FG} = \frac{FJ}{FJ} \)  – Definition of Similarity  
6. \( FH = FG + GH \)  – Additive Property of Length  
7. \( FI = FJ + IJ \)  – Additive Property of Length  
8. Substituting into the ratio: \( \frac{FG + GH}{FG} = \frac{FJ + IJ}{FJ} \)  
9. Properties of addition, subtraction, multiplication, and division are applied.

### Completing the proof:

To finalize, we simplify the equation in step 8 using the properties of addition:

\[
\frac{FG + GH}{FG} = 1 + \frac{GH}{FG}
\]
\[
\frac{FJ + IJ}{FJ} = 1 + \frac{IJ}{FJ}
\]

Thus, the equation becomes:

\[
1 + \frac{GH}{FG} = 1 + \frac{IJ}{FJ}
\]

By subtracting 1 from both sides, we obtain:

\[
\frac{GH}{FG} = \frac{IJ}{FJ}
\]

This completes the proof!

Would you like any further details or clarifications? Here are some follow-up questions to consider:

1. What is the significance of using the AA similarity criterion in this proof?
2. How do corresponding angles play a role in proving triangle similarity?
3. Why is the reflexive property of congruence useful in geometric proofs?
4. Can you explain how the additive property of length was applied in this proof?
5. What other theorems could have been used to prove the similarity of the triangles?

**Tip:** Always start geometric proofs by identifying any congruent or similar triangles—it often simplifies the steps ahead.

Explore the step-by-step proof demonstrating that GH/FG equals IJ/FJ, utilizing concepts of similar triangles, corresponding angles, and properties of proportions. This guide is perfect for high school students in geometry classes.

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