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)
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.
To prove Theorem 3-8, we will use the following steps and justifications:
1. Given: r is perpendicular to s, and t is perpendicular to s. (Given)
2. Assume for contradiction: r is not parallel to t.
3. Since r is perpendicular to s, and t is perpendicular to s, angle 1 is a right angle and angle 5 is a right angle. (Definition of perpendicular lines)
4. Since angle 1 is a right angle and angle 5 is a right angle, angle 1 and angle 5 are congruent. (Definition of right angles)
5. Let's consider the two lines, r and t. If they are not parallel, then there must be another line, call it u, that intersects both r and t at different points. (Assumption for the contradiction)
6. Since line u intersects r and s, it forms angle 2 and angle 3. Similarly, it forms angle 6 and angle 7 when intersecting with t and s, respectively.
7. Since angles 2 and 3 are formed by line u and r, and angles 6 and 7 are formed by line u and t, we can conclude that angle 2 is congruent to angle 6 and angle 3 is congruent to angle 7. (Vertical angles)
8. Since angle 2 is congruent to angle 6 and angle 3 is congruent to angle 7, it follows that angle 2 is congruent to angle 7. (Transitive property of congruence)
9. Since angle 2 is congruent to angle 7, it means that angle 1 plus angle 2 is congruent to angle 1 plus angle 7. (Addition property of equality)
10. Angle 1 plus angle 2 is equal to 90 degrees, since they form a straight angle with right angle angle 1. Therefore, angle 1 plus angle 7 is also equal to 90 degrees.
11. Since angle 1 plus angle 7 equals 90 degrees and angle 5 is a right angle, angles 1 plus angle 7 is congruent to angle 5. (Substitution property of equality)
12. Since angle 5 is congruent to angle 1 plus angle 7, angle 5 is also congruent to angle 1 plus angle 2. (Transitive property of congruence)
13. Since angle 5 is congruent to angle 1 plus angle 2, and angle 5 is a right angle, it follows that angle 1 plus angle 2 is also equal to 90 degrees. (Substitution property of equality)
14. However, we know that angle 1 plus angle 2 is equal to 90 degrees, which contradicts our assumption that r is not parallel to t. Therefore, our assumption was incorrect.
15. Thus, r must be parallel to t. (Proof by contradiction)
16. Therefore, Theorem 3-8 is proved. (Conclusion)