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61 Angles in Polygons.notebook October 05, 2015 61: <s in Polygons Remember the Triangle Sum Theorem? It said the interior angles of a triangle add up to 180 degrees. We will use this later and apply it to polygons! We name polygons based on their number of sides: n=3 n=4 triangle quadrilateral n=7 n=8 heptagon octagon n=5 n=6 pentagon hexagon n=9 n=10 nonagon decagon 61 Angles in Polygons.notebook October 05, 2015 and... More than 12? Call it an "ngon" n = 12 dodecagon ex) 15 sides would be called a 15gon Diagonals: segments drawn from one vertex of a polygon to another. (must be drawn to non adjacent vertices) 61 Angles in Polygons.notebook October 05, 2015 So, how do we determine the sum of the interior angles of a polygon? Start by... Drawing all possible diagonals from one vertex of the polygon. By drawing non intersecting diagonals. These diagonals make triangles inside of polygons. n = 4 n = 5 n = 6 n = 7 n = 8 n = 9 RULE: Number of triangles = n 2 What we find is that the number of triangles inside of a polygon relates to the number of sides. # triangles = n 2 So, if we know that a triangle has 180 degrees and we know how many triangles are inside of a polygon, we can determine how many degrees are inside by multiplying (n 2)180. 61 Angles in Polygons.notebook Sum interior angles of a polygon can be found by S = 180(n 2) n 2 tells us how many triangles can be drawn with non intersecting diagonals 180 is how many degrees the angles of a triangle add up to ex) Find the sum of the interior angles of a hexagon. What is n? October 05, 2015 61 Angles in Polygons.notebook October 05, 2015 ex] Find missing angles: 5x (11x+4) (11x+4) 5x First, we will need to find the sum of the interior angles. Then, we will need to solve for x. Finally, we can plug in for x and find each angle measure. Regular Polygons All sides are congruent All int angles are congruent Our first example of this was an equiangular, equilateral triangle Triangle: n = 3 180(n 2) = 180 180 / 3 = 60o 61 Angles in Polygons.notebook October 05, 2015 To find the measure of an interior angle of a regular polygon, take the sum and divide it by the number of angles. Measure of the int angles of a regular polygon: 180(n 2) n ex) Find the measure of an interior angle of a regular pentagon. What is n? Then, plug into formula 61 Angles in Polygons.notebook October 05, 2015 ex) Find the measure of an interior angle of a regular Octagon. WHAT IS n??? ex) An interior angle of a regular polygon measures 144 degrees. How many sides does the polygon have? 61 Angles in Polygons.notebook October 05, 2015 ex) A regular polygon has an angle measuring 140 degrees. Find the number of sides of the polygon. Exterior Angles: All exterior angles of a polygon sum to 360, regardless of how many sides the polygon has. more below In a regular polygon the exterior angles are congruent. So, for a regular polygon you can find the measure of a single exterior angle by dividing 360 by n. Exterior Angles = 360 n 61 Angles in Polygons.notebook October 05, 2015 ex) Find the sum of the exterior angles of a dodecagon. ex) Find the measure of an exterior angle of a regular pentagon. Recap: What is n? Sum of interior angles of a polygon: Measure of interior angles of regular polygon: Sum of exterior angles of a polygon: Measure of exterior angles of regular polygon: 61 Angles in Polygons.notebook Homework: pg 397 1 23 October 05, 2015