Find the acute angle between the lines whose slopes are 2 & 1\3.

30 degrees

Step-by-step explanation:There are a couple of ways to do this.

This is the straightforward, “follow-your-nose” way. Find a point on each line. (Say, (1,2) on the line with slope 2 and (-1,1) on the line with slope -1. Then the angle in the triangle formed by the origin (0,0) and these other two point is the desired angle. One can find the cosine of angle in the triangle using the Law of Cosines, since the lengths of all of the sides can be found directly from the distance formula and the coordinates of the points. Finally, use arccosine to get from the cosine of the angle to the angle itself.

There is also a “slick” way, which relies on remembering the subtraction formula for tangents. tan(u-v) = [tan(u) - tan(v)] / [tan(u) + tan(v)]. Consider u as the angle between the positive x-axis and the line with slope -1, and v as the angle between the positive x-axis and the line with slope 2. Then the requested angle is u-v. We know tan(u) = -1 and tan(v) = 2, so substituting into the subtraction formula for tangents allows us to find tan(u-v). Finally, use arctangent of that value to find u-v.

ryan had a head start of 10 meters.