We are asked to show that if two nonvertical lines have equal slopes, then they are parallel. Let's consider the given diagram and focus on △ ABC and △ DEF. First we will show that these two triangles are similar.
The slope of a line is given as the ratio of the rise to the run between any two points on the line.
slope=rise/run
Let's use the points on the diagram to write an expression for the slopes of the lines.
Line
Points
Slope=rise/run
l_1
A,B
BC/AC
l_2
D,E
EF/DF
It is given that the slopes are equal.
BC/AC=EF/DF
Let's rearrange the above equation a little bit by using the Multiplication Property of Equality.
The above tells us that the ratio of the two legs of the right triangles △ ABC and △ DEF are the same. Since right angles are congruent, we also know that these triangles have a pair of congruent included angles. By the Side-Angle-Side Similarity Theorem, the triangles are similar.
△ ABC~ △ DEF
Corresponding angles in similar triangles are congruent.
∠ BAC≅∠ EDF
Let's mark these congruent angles on the diagram and focus on the x-axis as a transversal to lines l_1 and l_2.
