Short Answer
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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)
In the given diagram, we have lines r, t and s. Given that r is perpendicular to s and t is perpendicular to s, we want to prove that r is parallel to t.
To prove this, we can use the alternate interior angles theorem.
By definition of perpendicular lines, angles 1 and 6 are right angles as they are formed by the intersection of r and s, and t and s respectively.
Since angles 1 and 6 are right angles, they are congruent.
According to the alternate interior angles theorem, if two lines are cut by a transversal such that alternate interior angles are congruent, then the lines are parallel.
In this case, angle 6 is an alternate interior angle to angles 3 and 4, which are also congruent to angle 6.
Therefore, by the alternate interior angles theorem, we can conclude that r is parallel to t.