Geometry A unit 4 lesson 32 "parallel and perpendicular lines apply" portfolio connexus - 70 POINTS ONLY IF COMPLETY ANSWERED 1. On a coordinate plane, plot points A, B, and C such that B is the only point AB¯¯¯¯¯ and BC¯¯¯¯¯ have in common and neither of those line segments is horizontal or vertical. Give and then follow the steps needed to construct the unique point D such that AD¯¯¯¯¯∥BC¯¯¯¯¯ and AB¯¯¯¯¯∥CD¯¯¯¯¯ . Find the slopes of those line segments, and explain how they confirm those pairs of line segments are parallel.
2. Pick two integers a and b and two positive integers p and q. On a coordinate plane, draw the rectangle with vertices A(a, b) , B(a+p, b) , C(a+p, b+q) , and D(a, b+q) . Describe what you could construct to find the points W and X that partition AB¯¯¯¯¯ and DC¯¯¯¯¯ , respectively, in 1:1 ratios, then perform that construction. Using any method, locate the points Y and Z that partition AD¯¯¯¯¯ and BC¯¯¯¯¯ , respectively, in the same ratio of your choice, making sure that YZ¯¯¯¯¯ is horizontal. Let T be the intersection point of WX¯¯¯¯¯¯ and YZ¯¯¯¯¯ . Find the area of ABCD. Then find the areas of AWTY, BZTW, CXTZ, and DYTX. Describe how the ratios in which the four sides are partitioned relates to the areas of the large rectangle and the four smaller rectangles.
3. Write 4-6 sentences describing how many acute and obtuse angles are created when 2, 3, or 4 parallel lines are crossed by a single transversal that is not perpendicular to them. Create pictures to help. Identify how many unique acute angle measures there are and how many unique obtuse angle measures there are, explaining your reasoning. Propose a generalization for when there are n parallel lines, and confirm the generalization matches what was found for the cases of 2, 3, or 4 parallel lines. Prove the generalization, explaining all of your reasoning.