https://file.aiquickdraw.com/mathbot/file/1725497149782ydlvvr5d.jpg

State if the triangles are similar. If so, state the postulate and the similarity statement.

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.

Learn how to determine if two triangles are similar using the Angle-Angle (AA) Similarity Postulate. This example shows how to analyze given angles and establish similarity between triangles STG and FHG, suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation