# Write a two column proof. Given: a ll b, a`_|_` c, and b`_|_` d Prove: c ll d Statements:                                  Reasons

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It is given that lines a and b are parallel. If the slope of line a is S, the slope of line b is also S.

Line c is perpendicular to a. The product of the slope of perpendicular lines is -1. The slope of line c is S' = -1/S. Similarly, as the line b is perpendicular to line d, the slope of line d is also S' = -1/S.

Two lines with the same slope are parallel. This is the case with lines c and d, hence they are parallel.

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Introduce angles 1, 2, 3 and 4 in the diagram such that angles 1 and 4 are right angles (see the attached image).

Statements:                   Reasons:

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1. a ll b                         1. Given

2. `/_1` =` /_2`                2. If two lines are parallel, then their                                           alternate interior angles are congruent.

3. `/_1 ` = 90°                 3. Given

4.` /_2` = 90°                  4. Transitive property

5. `/_4` = 90°                 5. Given

6. `/_3` = 180-90=90°    6. Linear pair of adjacent angles are

supplementary.

7. `/_2` = `/_3`               7. All right angles are equal.

8. c ll d                       8. If a pair of alternate interior angles are

congruent, then the lines are parallel.

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