https://file.aiquickdraw.com/mathbot/file/1728768559892xkf6jcw6.jpg

To prove that \(\triangle RST \cong \triangle VUT\) using the given information, we can follow a standard geometric proof, relying on congruence postulates like SAS (Side-Angle-Side) or SSS (Side-Side-Side). Here's the step-by-step approach based on the image:

### Given Information:
- \( \overline{ST} \cong \overline{UT} \) (given)
- \( \overline{RT} \cong \overline{VT} \) (given)
  
We need to prove that \(\triangle RST \cong \triangle VUT\).

### Plan of Proof:
We will use **SAS** (Side-Angle-Side) congruence. Here's the reasoning:
1. **Side 1**: \( \overline{ST} \cong \overline{UT} \) (given).
2. **Side 2**: \( \overline{RT} \cong \overline{VT} \) (given).
3. **Angle \(T\)**: \( \angle RTS = \angle VUT \) (This is the vertical angle, and vertical angles are congruent).

### Complete Proof:
| **Statement**                             | **Reason**                               |
|-------------------------------------------|------------------------------------------|
| 1. \( \overline{ST} \cong \overline{UT} \) | Given                                    |
| 2. \( \overline{RT} \cong \overline{VT} \) | Given                                    |
| 3. \( \angle RTS \cong \angle VUT \)      | Vertical angles are congruent            |
| 4. \( \triangle RST \cong \triangle VUT \) | SAS Congruence Postulate (Side-Angle-Side) |

This proof shows that the triangles are congruent using the SAS postulate, which requires two pairs of corresponding sides and the included angle to be congruent.

Do you have any specific questions or would like further details?

### Related Questions:
1. How can we prove congruence using other postulates such as ASA (Angle-Side-Angle)?
2. What are vertical angles, and why are they congruent?
3. How does the SAS postulate differ from the SSS postulate?
4. Can the triangles be proven congruent using only sides (SSS postulate)?
5. What are other common ways to prove triangle congruence?

**Tip**: Always ensure that the angle in question is the one included between the two sides when applying the SAS postulate.

Use the given information to prove that ΔRST ≅ ΔVUT.

Learn how to prove that triangles RST and VUT are congruent using the SAS (Side-Angle-Side) congruence postulate. This problem involves understanding the properties of vertical angles and applying geometric theorems, suitable for students in Grades 9-10.

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