write a paragraph proof of theorem 3-8: if two lines are perpendicular to the same line then they are parallel to each other
Given: Line l, Line m and Line n such that Line l is perpendicular to Line n and Line m is perpendicular to Line n.
To Prove: Line l is parallel to Line m.
Proof:
1. Assume, to the contrary, that Line l and Line m are not parallel to each other.
2. If two lines are not parallel, they will eventually intersect.
3. Let Point A be the point of intersection between Line l and Line m.
4. Since Line l is perpendicular to Line n, the angle formed between Line l and Line n is a right angle.
5. Similarly, since Line m is perpendicular to Line n, the angle formed between Line m and Line n is a right angle.
6. Angles formed by perpendicular lines intersecting with a transversal (Line n) are congruent.
7. Therefore, the angle formed between Line l and Line m at Point A is a right angle.
8. But this contradicts the assumption made in step 1 that Line l and Line m are not parallel to each other.
9. Thus, our assumption was incorrect, and Line l must be parallel to Line m.
Therefore, if two lines are perpendicular to the same line, then they are parallel to each other.