Question

I need help on proving that nonvertical lines are parallel if and only if they have the same slop.

Answer 1

So,

We could prove this statement as follows:

First let's draw two nonvertical parallel lines. These are:

If we draw two perpendicular lines, we will obtain two similar triangles:

We can notice that:

`\Delta AEB\approx\Delta DEC`

Which means that triangles AEB and DEC are similar by AAA statement.

If any two angles of a triangle are equal to any two angles of another triangle, then the two triangles are similar to each other. (This is the Angle-Angle similarity).

Now, as these triangles are similar, we could say that:

`(BE)/(AE)=(CE)/(DE)`

If you see, BE/AE is the slope of line AB and CE/DE is the slope of line CD. Now, we have proved that the slope of lines AB and CD are equal, thus nonvertical lines are parallel if and only if they have the same slope.