 # Classifying angles

Connect grid points with straight lines to form a path. Your goal is to create as many acute angles as possible, whether those angle are interior to some polygon or not. The path, whether open or closed, cannot intersect itself, nor come into contact with itself. Here's one solution: After students have found the solution with 14 acute angles, or have been shown the answer after struggling significantly, ask them these further questions: For which other wxh grids can you find wxh - 2 acute angles? For the grids where wxh - 2 acute angles is impossible, how many angles can you find in each case?

Problem:

Given n lines, what is the maximum number of acute angles you can make? We know every angle is comprised of two rays, and thus, we need at least 2 lines to get at least 1 angle. Given n lines, which number of acute angles can you make? For example, given 2 lines, you can make 0 acute angles, by setting the lines parallel or perpendicular, or you can make 2 acute angles, by having the lines intersect without being perpendicular, but you cannot make any other number. What about using n lines to construct other types of angles: right, obtuse, reflex?