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Prove and use slope criteria for parallel and perpendicular lines.

Decide whether two lines are parallel by comparing their slopes

Problem
Are two distinct lines with slopes 3 and 3 parallel?
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Hint

Compare the two slopes directly.

Lines are parallel when they have equal slopes.

Solution walkthrough

01

Recall the parallel-line criterion

\[\text{distinct}~\text{nonvertical}~\text{lines}~\text{are}~\text{parallel}~\text{when}~\text{their}~\text{slopes}~\text{are}~\text{equal}\]

Equal slope means equal direction; the prompt explicitly says the lines are distinct.

02

Compare the slopes

\[3=3\]

The two supplied slopes are identical.

03

State the relationship

\[\text{Yes},~\text{they}~\text{are}~\text{parallel}.\]

Because the lines are distinct and have equal slopes, the parallel conclusion follows.

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Another way

  1. Both lines rise 3 units for every 1 unit of run, so they have the same direction.

!

Common mistake

Do not reject the lines because both slopes are positive. The sign does not matter for parallelism; equality does.