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Category 2 Geometry Meet #1, October 2012 1. In the diagram below, lines k and j are parallel. The measure of angle IHJ is 20 degrees and the measure of angle DCH is 80 degrees. How many degrees are there in the measure of angle CGH? 2. The supplement of a certain angle is 5 more than 6 times the angle itself. How many degrees are there in the measure of the complement of this angle? 3. In the figure below, two squares and three equilateral triangles are constructed on the sides of irregular pentagon IKMPR. How many degrees are there in the sum of the measures of angles HIJ, JKL, LMN, OPQ, and QRS? Answers 1. ______________degrees 2. ______________degrees 3. ______________degrees Solutions to Category 2 Geometry Meet #1, October 2012 1. Angle CHG = 20 degrees (vertical angles). Angle GCH = 100 degrees (supplementary angles). Thus angle CGH = 180 – 100 – 20 = 60 degrees. Answers 1. 60 degrees 2. 65 degrees 3. 540 degrees 2. Translating the words to algebra, we can write the equation 180 – x = 6x + 5. This simplifies to 7x = 175, so x = 25. The complement of x is therefore 90 – 25 = 65 degrees. 3. There is a total of 5 360 = 1800 degrees about points I, K, M, P, and R. Since a pentagon can be divided into three triangles, the total interior angle sum of any pentagon is 3 180 = 540. Whatever the measures of the particular interior angles of pentagon IKMPR, this total of 540 degrees can be subtracted from our total of 1800, leaving 1800 – 540 = 1260 degrees. We will also subtract four 90-degree angles for the two squares and six 60-degree angles for the triangles. That leaves 1260 – 4 90 – 6 60 = 1260 – 360 – 360 = 900 – 360 = 540 degrees.