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Let's break down the proof in steps and fill in the reasoning to demonstrate that \(\overrightarrow{UW} \perp \overrightarrow{TV}\).

### Step-by-Step Completion:

1. **Statement: \(\angle TXU \cong \angle UXV\)**
   - **Reason: Vertical angles are congruent.**
     - Explanation: Since the two angles are opposite each other at point \(X\), they are vertical angles and thus congruent.

2. **Statement: \(m\angle TXU + m\angle UXV = 180^\circ\)**
   - **Reason: Linear pair of angles.**
     - Explanation: \( \angle TXU \) and \( \angle UXV \) form a straight line, which means their measures add up to 180 degrees (linear pair postulate).

3. **Statement: \(m\angle TXU + m\angle TXU = 180^\circ\)**
   - **Reason: Substitution.**
     - Explanation: Since \(\angle TXU \cong \angle UXV\), the measure of \(\angle UXV\) can be substituted with \(m\angle TXU\), so we add the same angle twice.

4. **Statement: \(m\angle TXU = 90^\circ\)**
   - **Reason: Solve for \(m\angle TXU\).**
     - Explanation: By solving the equation \(m\angle TXU + m\angle TXU = 180^\circ\), you get \(m\angle TXU = 90^\circ\), meaning the angle at \(X\) is a right angle.

5. **Statement: \(\overrightarrow{UW} \perp \overrightarrow{TV}\)**
   - **Reason: Definition of perpendicular lines.**
     - Explanation: Since \(\angle TXU = 90^\circ\), the two vectors \(\overrightarrow{UW}\) and \(\overrightarrow{TV}\) are perpendicular by the definition of perpendicularity (two lines that meet at a right angle).

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Would you like further clarification or details on any of the steps?

Here are five related questions for further exploration:
1. What other geometric properties are linked to vertical angles?
2. How can the concept of perpendicularity be applied to vector algebra?
3. What is the significance of a linear pair in geometric proofs?
4. How do we prove the perpendicularity of two lines using coordinate geometry?
5. How can angle bisectors be used in similar proofs?

**Tip:** In any geometric proof involving perpendicularity, always check for right angles or supplementary angles formed by intersecting lines.

Complete the proof that vector UW is perpendicular to vector TV.

Learn how to complete a geometry proof that demonstrates the perpendicularity of vectors UW and TV. This problem involves key geometric concepts such as congruent angles, the linear pair postulate, and the definition of perpendicular lines. Suitable for high school students in Grades 9-10.

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