We are asked to show that nonvertical parallel lines have equal slopes. Let's focus on △ ABC and △ DEF.
First we will show that these two triangles are similar.
Let's see what we can deduce from the fact that lines l_1 and l_2 are parallel.
The x-axis is a transversal that intersects two parallel lines. Therefore, by the Corresponding Angles Theorem, we know that l_1 and l_2 intercept the x-axis at congruent angles. Moreover, BC and EF are both perpendicular to the x-axis. Since right angles are congruent, we have a second pair of congruent angles.
∠ BAC≅∠ EDF
and
∠ BCA≅∠ EFD
Let's show this information in our diagram.
We now see that two angles of △ ABC are congruent to two angles of △ DEF.
By the Angle-Angle Similarity Postulate, we can then conclude that the two triangles are similar.
△ ABC~ △ DEF
We know that corresponding sides of similar triangles are proportional.
BC/EF=AC/DF
Let's rearrange the above proportion to obtain the desired equation using the Multiplication Property of Equality.
Note that BCAC and EFDF are the slopes of l_1 and l_2, respectively. Therefore, we have proved that the slopes of nonvertical parallel lines are the same. We can summarize the steps above in a flow proof.
