SOLUTION: derive the condition for two lines with slope m1 and m2 to be parallel and perpendicular using the angel between two lines formula.

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Question 1203322: derive the condition for two lines with slope m1 and m2 to be parallel and perpendicular using the angel between two lines formula.
Answer by math_tutor2020(3817) About Me  (Show Source):
You can put this solution on YOUR website!

theta = theta = angle between two lines

One formula to use is
tan%28theta%29+=+abs%28%28m%5B1%5D-m%5B2%5D%29%2F%281+%2B+m%5B1%5Dm%5B2%5D%29%29

Parallel lines will have theta = 0.
This leads to tan(theta) = tan(0) = 0.
The right hand side is only zero when the numerator is zero.
m%5B1%5D-m%5B2%5D+=+0 which rearranges to m%5B1%5D+=+m%5B2%5D
Therefore, parallel lines have equal slopes.
For example, the lines y = 3x+5 and y = 3x+7 are parallel. Each has slope of 3.

Perpendicular lines will involve theta = 90 degrees.
Use a unit circle to determine that tan(90) is undefined.
When it comes to "undefined", it means we have 0 in the denominator.
Division by zero is not allowed.
1+%2B+m%5B1%5Dm%5B2%5D+=+0 leads to m%5B2%5D+=+-1%2F%28m%5B1%5D%29 which means we take the negative reciprocal of m1 to get m2, and vice versa.
An example pair of perpendicular slopes: m%5B1%5D+=+5%2F7 and m%5B2%5D+=+-7%2F5
Note that any pair of perpendicular slopes multiply to -1. Neither slope can be zero or undefined.