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The proof presented in the image asks to demonstrate that horizontal lines are perpendicular to vertical lines. Here is a step-by-step explanation with the missing blanks filled in:

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### Proof: Prove that horizontal lines are perpendicular to vertical lines.

**By definition,** the \( x \)-axis is **perpendicular** to the \( y \)-axis. Let \( m \) be a horizontal line, and let \( n \) be a vertical line. 

- Because any two **vertical lines** are parallel, \( m \) is parallel to the **x-axis**.
- Because any two **horizontal lines** are parallel, \( n \) is parallel to the **y-axis**.

Then, by the **Perpendicular Transversal Theorem**, \( n \) is perpendicular to \( m \).

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So, the blanks are filled as follows:

1. **perpendicular** 
2. **y-axis**
3. **vertical lines**
4. **x-axis**
5. **Perpendicular Transversal Theorem**

Would you like further details on this theorem or clarification on any specific part of the proof?

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### Related Questions:
1. What is the Perpendicular Transversal Theorem?
2. How can you prove that two lines are parallel?
3. Why are the x-axis and y-axis always perpendicular by definition?
4. Can a line be perpendicular to more than one line at a point?
5. How does the definition of perpendicular lines apply in non-Euclidean geometries?

#### Tip:
When studying geometric theorems, it's helpful to sketch diagrams to visualize the relationships between lines and angles, reinforcing your understanding of how perpendicularity and parallelism are established.

Prove that horizontal lines are perpendicular to vertical lines.

This page presents a detailed proof showing that horizontal lines are perpendicular to vertical lines using the Perpendicular Transversal Theorem. It includes key concepts such as the definition of perpendicular and parallel lines in geometry. Suitable for Grades 8-10.

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