https://file.aiquickdraw.com/mathbot/file/1726483550917by7ufhro.jpg

Let's examine the problem from the image and break it down to determine what additional information is needed for congruence.

### 1. Problem 1: SAS (Side-Angle-Side) Congruence

You have two triangles \( \triangle UWV \) and \( \triangle KVM \) that are partially given.

- Side \( UW = KV \) (marked with a single tick).
- Side \( VM = VK \) (marked with a double tick).

**Additional Information Required**:  
To prove congruence using the SAS postulate, we need to know that the angle between these two sides is congruent in both triangles. The angle between sides \( UW \) and \( VM \) in \( \triangle UWV \) should be congruent to the angle between sides \( KV \) and \( VM \) in \( \triangle KVM \).

- **Required angle**: \( \angle UVW \cong \angle KVM \).

**Congruence Statement**:  
If this angle information is provided, the triangles would be congruent by SAS, and we can write the congruence statement as:
\[ \triangle UWV \cong \triangle KVM \]

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### 2. Problem 2: ASA (Angle-Side-Angle) Congruence

In this case, you have two triangles \( \triangle UDS \) and \( \triangle TDS \).

- \( \angle UDS \) is marked congruent to \( \angle TDS \).
- Side \( DS \) is shared between the two triangles, so it's congruent by definition (a common side).

**Additional Information Required**:  
To apply the ASA postulate, we need one more pair of angles to be congruent. Specifically, we need \( \angle USD \) to be congruent to \( \angle TSD \) (the angles opposite the side \( DS \)).

- **Required angle**: \( \angle USD \cong \angle TSD \).

**Congruence Statement**:  
Once this angle information is provided, the triangles would be congruent by ASA, and we can write the congruence statement as:
\[ \triangle UDS \cong \triangle TDS \]

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Do you have any questions or need further clarification? 

Here are five related questions you might want to explore:
1. What other congruence criteria can be used to prove triangles are congruent besides SAS and ASA?
2. Why does the angle need to be between the two sides for the SAS postulate?
3. What is the importance of a common side when proving triangle congruence?
4. Can we prove triangle congruence if only two angles are given but not a side (i.e., just AA)?
5. How can the SSS (Side-Side-Side) congruence postulate be applied to other triangles?

**Tip**: When using postulates like SAS and ASA, always check if the angle is between the two sides (for SAS) or if the two angles are around the known side (for ASA).

State what additional information is required in order to know that the triangles are congruent for the reason given. Then write the congruence statement for SAS and ASA cases.

Learn how to determine the congruence of triangles using the SAS (Side-Angle-Side) and ASA (Angle-Side-Angle) postulates. This problem covers key geometric concepts, providing examples and explanations for both congruence criteria.

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