To determine if the triangles $\triangle STG$ and $\triangle FHG$ are similar, let's analyze the information provided in the image.

Given Data:

• $\angle STG = 63^\circ$
• $\angle GHF = 61^\circ$

We need to check if the two triangles share any of the known similarity postulates (AA, SSS, or SAS):

Step 1: Identify the Angles

1. In $\triangle STG$, one angle $\angle STG = 63^\circ$ is given.
2. In $\triangle FHG$, one angle $\angle GHF = 61^\circ$ is given.

Step 2: Check the Relationship Between Angles

To determine if the triangles are similar by the AA (Angle-Angle) postulate, we need to verify if the angles in $\triangle STG$ and $\triangle FHG$ are congruent.

1. $\angle HGT$ is a vertical angle with $\angle STG$. Therefore, $\angle HGT = 63^\circ$.
2. The sum of angles in $\triangle FHG$ must be $180^\circ$. Thus, the angle $\angle HGF$ can be found as: $\angle HGF = 180^\circ - 61^\circ - 63^\circ = 56^\circ$

Step 3: Conclude the Similarity

• $\triangle STG$ and $\triangle FHG$ both have angles $63^\circ$ and $61^\circ$.
• This gives us two pairs of congruent angles.

Conclusion:

Since $\triangle STG$ and $\triangle FHG$ have two corresponding angles that are congruent, the triangles are similar by the AA (Angle-Angle) Similarity Postulate.

Similarity Statement:

$\triangle STG \sim \triangle FHG$

Would you like more details or have any questions?

Further Related Questions:

1. How can we use the SSS (Side-Side-Side) postulate to determine similarity?
2. What are some real-world applications of triangle similarity?
3. How does the SAS (Side-Angle-Side) postulate work in establishing similarity?
4. Can non-similar triangles have corresponding congruent angles? Why or why not?
5. What is the importance of the AA postulate in geometry?

Tip:

Always check if any angles are vertically opposite or if they sum to 180 degrees when trying to prove similarity.