Unit 5, Lesson 5
Parallel Lines in the Plane
Integrated Math 1
5.1
Warm-up
- Draw any non-vertical line in the plane. Draw 2 possible translations of the line.
- Find the slope of your original line and the slopes of the images.
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Priya writes a proof saying:
Consider any 2 parallel lines. Assume they are not horizontal or vertical. Construct a horizontal line that crosses both of the parallel lines. Then construct a vertical line through the midpoint of the horizontal segment between the lines. This forms 2 congruent triangles. Since the sides of the right triangles are horizontal and vertical, we can use them to find the slope of the parallel lines. Therefore the parallel lines have the same slope.
How does Priya know that the right triangles are congruent?
The solid line has been translated in 2 different ways:
The 3 lines look parallel to one another, as we would expect. We know that translations of lines result in parallel lines.
What happens to the slopes of these lines? If we draw in the slope triangles that go through the origin, we can see right triangles. Since we know the lines are parallel, the corresponding pairs of angles in the triangles must be congruent by the Alternate Interior Angles Theorem. Triangles with congruent angles are similar, and similar slope triangles result in lines with the same slope. Here we see slopes of and , which are all equal.
Coordinate plane. Heavy dotted line through 0 comma 6, 2 comma 4. Solid line through 0 comma 3, 2 comma 2, shaded blue below the line and in the first quadrant. Light dotted line through 0 comma negative 1, 2 comma negative 2, shaded yellow above this line and in the third quadrant.
We can use similar reasoning to show that any 2 parallel lines that aren’t vertical have the same slope, and also that any 2 lines with the same slope are parallel.
What if we wanted to find the equation of a line parallel to these 3 lines that goes through the point ? We know the line must have the same slope of . We can use point-slope form and get .