Write a paragraph proof of theorem 3-8: in a plane, if two lines are perpendicular to the same line, then they are parallel to each other.
so far all i have is this
r and t are each perpendicular to s. Angles 1 & 5 have to be right angels.
but my teacher says its not enough details can anyone help me??
If you guys can help me i can give the answers to the test! This is the only question im stuck on!!
To prove theorem 3-8, we need to show that if two lines, say r and t, are perpendicular to the same line s, then they are parallel to each other. Here is a step-by-step paragraph proof:
1. Given: Line r is perpendicular to line s, and line t is perpendicular to line s.
2. We want to prove: r is parallel to t.
3. Assume, for the sake of contradiction, that r and t are not parallel. This means that they intersect at some point, say point P.
4. Since r is perpendicular to s, angle 1 (formed by lines r and s) is a right angle.
5. Similarly, since t is perpendicular to s, angle 5 (formed by lines t and s) is also a right angle.
6. If r and t intersect at P, then angles 1 and 5 are congruent (by vertical angles theorem).
7. But both angles 1 and 5 are right angles, so they are congruent.
8. Therefore, we have a contradiction, as angles 1 and 5 cannot be both congruent and right angles if r and t are not parallel.
9. Hence, our assumption that r and t are not parallel is false.
10. Therefore, r and t must be parallel to each other.
By following this step-by-step proof, you can demonstrate that if two lines are perpendicular to the same line, then they must be parallel to each other.