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Suppose we have a line such that this line and the positive -axis makes an angle .

As we can see the slope of the line is . Now if the line is going downward as in this picture

then the slope of is .

Now we are ready to prove that if are slopes of two lines, then if and only if the two lines are perpendicular. First look at the following picture

Let be the slopes of respectively. We want to show that . By the previous discussion this equivalent to
.

Hence if and only if if and only if if and only if the two lines are perpendicular.

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