https://file.aiquickdraw.com/mathbot/file/1725495924755v3agh0n3.jpg

Type similarity postulate, then letters that represent the verticles of the similar triangle, where applicable..

From the image, we can see a triangle labeled \( \triangle FGH \) with the side lengths:

- \( FG = 11 \)
- \( GH = 11 \)
- \( HP = 7 \)
- \( FP = 19 \)
- \( SP = 7 \) (length between points H and P, based on the marked 7 units).

It seems the question asks for the similarity postulate and the vertices of the triangles that are similar.

To determine triangle similarity, we usually use one of the following postulates:

1. **AA (Angle-Angle) Similarity**: If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar.
2. **SSS (Side-Side-Side) Similarity**: If the corresponding sides of two triangles are proportional, the triangles are similar.
3. **SAS (Side-Angle-Side) Similarity**: If two sides of one triangle are proportional to two sides of another triangle and the included angles are congruent, the triangles are similar.

### Identifying Similarity
Given the triangle \( \triangle FGH \) and the smaller triangle that seems to be \( \triangle HSP \), we observe that:
- Both triangles share angle \( H \), which could help in identifying similarity using the AA or SAS postulate.
- We also see proportionality in the side lengths: \( HP : FP \) and \( GH : FG \).

This suggests that the smaller triangle \( \triangle HSP \) is similar to \( \triangle FGH \) by the **SAS similarity postulate** (Side-Angle-Side).

Thus, the similar triangles are:

- \( \triangle FGH \sim \triangle HSP \)

Let me know if you'd like further clarifications or more details!

### Relative Questions:
1. What other methods can be used to verify triangle similarity?
2. How does the AA similarity postulate differ from SAS?
3. Can two triangles with different perimeters still be similar?
4. How does triangle congruence differ from triangle similarity?
5. What role does proportionality of side lengths play in determining similarity?

**Tip:** Always check for shared angles when using the AA postulate to determine triangle similarity.

Type similarity postulate, then letters that represent the vertices of the similar triangle, where applicable.

Understand triangle similarity postulates using the example of triangles FGH and HSP. This problem applies the SAS similarity postulate to determine the proportionality of the sides and angles of the triangles. Suitable for students in Grades 9-10.

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