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5.7: Sums of Angle Measures in Polygons Brinkman SKILLS TO AQUIRE οΌ Use the Triangle-, Quadrilateral-, and Polygon-Sum Theorems to determine angle measures. οΌ Know history and impact postulates had on parallel lines and the development of geometry. In the picture, connect points A, B, and C on the cheese head at the right. Fill in the blanks below with the correct angle measurement. A B πβ π΅π΄πΆ 98 πβ πΆπ΅π΄ 35 πβ π΅πΆπ΄ 47 Find πβ π΅π΄πΆ + πβ πΆπ΄π΅ + πβ π΅πΆπ΄ = πππ! Sweetβ¦ C Definition: Triangle β Sum Theorem The sum of the measures of the angles of a triangle is 180o. Prove this is true! Okayβ¦ Given: β³ π¨π©πͺ Prove: πβ π΄ + πβ π΅ + πβ πΆ = 180 a.) Draw line through A that is parallel to π΅πΆ . (Put protractor on π΄π΅ and point A, measure 90 degrees) b.) On line just drawn, put points P and Q on opposite sides of A, and where Q and C are on the same side of line AB. c.) What do lines AB and AC act as now? AB is the perp. Bisector of BC and AC is a transversal d.) Therefore πβ ππ΄π΅ = πβ π πππ πβ ππ΄πΆ = πβ π e.) We know that the angle measure of a line is 180o f.) Therefore πβ π΄ + πβ π΅ + πβ πΆ = 180! Practice! Laura Bora was working on her triangle sum homework and noticed that in triangle ABC, the angles are in the extended ration 1:2:3. This means for some real number x, the angles have measures 1x, 2x, and 3x. Find their measures! 2x B 1x + 2x + 3x = 180 6x = 180 x = 30 πβ π¨= 30 1x 3x A πβ π© = 60 C πβ πͺ = 90 When the Grinch wasnβt stealing Christmas, he was trying to figure out the sum of a quadrilateral. He was stumped, because math wasnβt really his thing. But then he had an βAhaβ moment! He noticed that if he took a Quadrilateral, he could turn this into 2 TRIANGLES! Oh happy day! He knew the sum of a triangle was 180o Since there are 2 TRIANGLES in the quadrilateral, he multiplied 2 x 180 and got 360o. Definition: Quadrilateral β Sum Theorem The sum of the measures of the angles of a quadrilateral is 360o. Notice that there are two triangles when there are 4 sides and 4 angles. Sums of Angles in Convex Polygons! Draw the number of triangles in each polygonβ¦ There are 3 triangles in a 5-gon. There are 4 triangles in a 6-gon. There are 5 triangles in a 7-gon. Angle measure: 540o Angle measure: 720 o Angle measure: 900 o Definition: Polygon β Sum Theorem The sum of the measures of the angles of a convex n-gon is 180(n β 2)o.