The problem asks to write a similarity statement relating the three triangles in the given diagram and to choose the correct option. Let’s break it down.

Solution:

Observations:

1. The triangle $\triangle EFG$ contains two right triangles:

   • $\triangle EHG$: Formed by point $H$ on $\overline{EF}$.
   • $\triangle GHF$: The smaller triangle where $\overline{GH}$ and $\overline{HF}$ meet.

2. Similarity:

   • All three triangles are similar by AA similarity:
     • They share one right angle ($\angle EHG, \angle GHF, \angle EFG$).
     • They all share an additional acute angle (e.g., $\angle GEF, \angle HEG, \angle FGH$).

Correct Option:

The similarity statement must align corresponding angles properly: $\triangle EFG \sim \triangle EHG \sim \triangle GHF$

Thus, the correct answer is:

• Third option: $\triangle EFG \sim \triangle EHG \sim \triangle GHF$.

Would you like a deeper explanation or a step-by-step geometry reasoning? Let me know! 😊

Related Questions:

1. Why are all three triangles similar by AA similarity?
2. What are the corresponding sides for the three triangles?
3. How can the proportionality of sides be used to solve for unknown lengths in similar triangles?
4. What is the geometric importance of the altitude $\overline{EH}$ in the larger triangle $\triangle EFG$?
5. How can you calculate angles in each triangle using similarity?

Tip:

When identifying similar triangles, always match corresponding angles to ensure proper order in the similarity statement!