Short Answer

Question

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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.

The information and diagram for a two column proof is shown.· Given colon r perpendicular s, t perpendicular s
· Prove colon r parallel t
· A diagram is shown of line r and line t with line s intersecting both line r and line t.
· Four angles are formed at the intersection of line r and line s. Angle 1 is in the top left position, angle 2 in the top right position, angle 3 in the bottom left position, and angle 4 in the bottom right position.
· Four angles are formed at the intersection of line t and line s. Angle 5 is in the top left position, angle 6 in the top right position, angle 7 in the bottom left position, and angle 8 in the bottom right position.
(4 points)

Answer 1

To prove that lines r and t are parallel, we can use the alternate interior angles theorem. In the given diagram, we can see that angles 1 and 6 are alternate interior angles formed by lines r and t and transversal line s. Since angles 1 and 6 are congruent and lines r and t are perpendicular to line s (given), we can conclude that lines r and t are parallel. This is because alternate interior angles are congruent if and only if the lines cut by the transversal are parallel. Thus, we have proved that in a plane, if two lines are perpendicular to the same line, then they are parallel to each other.