This exercise outlines a proof of the fact that two non-vertical lines are parallel if and...

This exercise outlines a proof of the fact that two non-vertical lines are parallel if and only if their slopes are equal. The proof relies on the following observation for the given figure: The lines y = m1x + b1 and y = m2x + b2 will be parallel if and only if the two vertical distances AB and CD are equal. (In the figure, the points C and D both have x-coordinate 1.)

(a) Verify that the coordinates of A, B, C, and D are A(0, b1) B(0, b2) C(1, m1 + b1) D(1, m2 + b2)

(b) Using the coordinates in part (a), check that AB = b1 − b2 and CD = (m1 + b1) − (m2 + b2)

(c) Use part (b) to show that the equation AB = CD is equivalent to m1 = m2.