In this exercise, we find a formula for the angle formed by two lines in a coordinate plane.
(a) If is a line in the plane and is the angle formed by the line and the -axis as shown in the figure, show that the slope of the line is given by
(b) Let and be two nonparallel lines in the plane with slopes and , respectively. Let be the acute angle formed by the two lines (see the following figure). Show that
(c) Find the acute angle formed by the two lines
and
(d) Show that if two lines are perpendicular, then the slope of one is the negative reciprocal of the slope of the other. [Hint: First find an expression for .]