To prove that lines \( SU \perp TV \), we need to show that these two lines are perpendicular. The approach depends on the given information or context (such as coordinate geometry, vectors, or geometric properties of the figure). Let's assume we're working with vectors, as that is a common way to prove perpendicularity.

### Proof using vectors:
Let \( S = (x_1, y_1) \), \( U = (x_2, y_2) \), \( T = (x_3, y_3) \), and \( V = (x_4, y_4) \) be the coordinates of points \( S \), \( U \), \( T \), and \( V \), respectively.

#### Step 1: Represent the lines as vectors
- The vector \( \overrightarrow{SU} \) representing line \( SU \) is:
  \[
  \overrightarrow{SU} = (x_2 - x_1, y_2 - y_1)
  \]
- The vector \( \overrightarrow{TV} \) representing line \( TV \) is:
  \[
  \overrightarrow{TV} = (x_4 - x_3, y_4 - y_3)
  \]

#### Step 2: Use the dot product to check for perpendicularity
Two vectors are perpendicular if and only if their dot product is zero. The dot product of \( \overrightarrow{SU} \) and \( \overrightarrow{TV} \) is:
\[
\overrightarrow{SU} \cdot \overrightarrow{TV} = (x_2 - x_1)(x_4 - x_3) + (y_2 - y_1)(y_4 - y_3)
\]

For \( SU \perp TV \), we need:
\[
(x_2 - x_1)(x_4 - x_3) + (y_2 - y_1)(y_4 - y_3) = 0
\]

If the above equation holds true, then the lines are perpendicular.

### Conclusion:
The lines \( SU \) and \( TV \) are perpendicular if the dot product of the vectors \( \overrightarrow{SU} \) and \( \overrightarrow{TV} \) equals zero.

Do you have any specific values for the points \( S, U, T, V \), or would you like to clarify more about the geometric context?

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**Would you like more details or have any questions?**

### 5 Related Questions:
1. How do you compute the dot product of two vectors?
2. What are the conditions for two vectors to be parallel instead of perpendicular?
3. How can you prove perpendicularity using slopes in coordinate geometry?
4. Can you prove perpendicularity in 3D space using vectors?
5. How do you use the cross product to determine if two vectors are perpendicular?

**Tip:** In coordinate geometry, the slopes of two perpendicular lines are negative reciprocals of each other.

Learn how to prove that lines SU and TV are perpendicular using vector representation and the dot product. This solution explores key concepts in coordinate geometry and vector algebra, including the dot product formula for perpendicularity. Suitable for high school students (Grades 9-12).

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