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: Line n is perpendicular to lines p and q
Prove: Line p is parallel to line q
Statement | Reason
1. Line n is perpendicular to line p | Given
2. Line n is perpendicular to line q | Given
3. ∠1 and ∠2 are right angles | Definition of perpendicular lines
4. ∠1 ≅ ∠2 | All right angles are congruent
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5. Line p is parallel to line q | If two angles are congruent and one pair of corresponding sides are concurrent, then the two lines are parallel (Alternate Interior Angles Theorem)
Therefore, we have proved that if two lines are perpendicular to the same line, then they are parallel to each other.
Statement | Reason
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1. Line AB is perpendicular to line CD. | Given
2. Line EF is perpendicular to line CD. | Given
3. Assume that lines AB and EF are not parallel. | Assumption
4. Let P be the intersection point of lines AB and EF. | Definition of intersection
5. Angle APC and angle EPD are right angles. | Definition of perpendicular lines
6. Angle BPC and angle EPD are right angles. | Definition of perpendicular lines
7. Angle BPC is congruent to angle APC. | All right angles are congruent
8. Angle APC is congruent to angle EPD. | All right angles are congruent
9. Angle BPC is congruent to angle EPD. | Transitive property of congruence
10. Triangles APC and EPD are congruent. | ASA congruence property
11. Line AP is congruent to line EP and line CP is congruent to line DP. | Definition of congruent triangles
12. Line CP is not congruent to line EP. | Contradiction (Assumption)
13. Line AB is parallel to line EF. | Contradiction (Assumption)
14. Assumption of lines AB and EF not being parallel was incorrect. | Contradiction
15. Therefore, lines AB and EF are parallel. | Law of Contradiction
16. Conclusion: If two lines are perpendicular to the same line, then they are parallel to each other. | Definition of parallel lines
This paragraph proof shows that if two lines, AB and EF, are perpendicular to the same line, CD, then they must also be parallel to each other. It begins by assuming that AB and EF are not parallel, and then uses the given information to show that this assumption leads to a contradiction. Thus, the assumption must be incorrect, and therefore, lines AB and EF must be parallel.