https://file.aiquickdraw.com/mathbot/file/1730471584929gc46z3fk.jpg

Determine whether the triangles are congruent.  If so, name the postulate or theorem that justifies your answer.  If not, explain.The triangles are congruent by the SAS Postulate.

The triangles are congruent by the AAS Postulate.

The triangles are congruent by the ASA Theorem.

The triangles are not necessarily congruent.  Not enough information provided.

To determine if the triangles in the image are congruent, let's analyze the given information and apply congruence postulates.

The image shows two triangles, \( \triangle UTS \) and \( \triangle RST \), which share side \( TS \). Here is a breakdown of the observations:

1. **Shared Side \( TS \)**: This side is common to both triangles, which gives us one pair of equal sides.
   
2. **Angle Markings**: The red arcs indicate congruent angles in both triangles, suggesting that \( \angle UTS \cong \angle RTS \).
   
3. **Second Pair of Angles**: Each triangle has another angle marked with red, likely indicating that \( \angle TUS \cong \angle TRS \).

### Conclusion
With two pairs of angles congruent and one side congruent (side \( TS \)), this fits the **ASA (Angle-Side-Angle)** postulate for triangle congruence.

### Answer
The triangles are congruent by the **ASA Theorem**.

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Do you want further details or have any questions? Here are some related questions to explore:

1. What is the ASA Theorem in detail?
2. How does the ASA Theorem differ from the SAS Postulate?
3. Can you explain how the AAS Postulate is used in triangle congruence?
4. What other conditions (postulates) can prove triangle congruence?
5. What happens if only two angles are congruent without any sides?

**Tip:** When analyzing triangle congruence, focus on identifying matching sides and angles carefully, as each postulate requires specific conditions.

Determine whether the triangles are congruent. If so, name the postulate or theorem that justifies your answer. If not, explain. The triangles are congruent by the SAS Postulate. The triangles are congruent by the AAS Postulate. The triangles are congruent by the ASA Theorem. The triangles are not necessarily congruent. Not enough information provided.

Learn how to determine triangle congruence with a step-by-step analysis of given triangles in a geometric figure. This guide explores congruence postulates such as ASA (Angle-Side-Angle), helping students understand triangle congruence criteria, suitable for Grades 8-10.

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