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Download 6.1 Polygons - Teacher Notes
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Transcript
3.5 The Polygon Angle-Sum Theorems Geometry Mr. Barnes Objectives: • To Classify Polygons • To find the sums of the measures of the interior and exterior angles of polygons. Q VERTEX R Definitions: SIDE P S VERTEX T • Polygon—a plane figure that meets the following conditions: – It is formed by 3 or more segments called sides, such that no two sides with a common endpoint are collinear. – Each side intersects exactly two other sides, one at each endpoint. • Vertex – each endpoint of a side. Plural is vertices. You can name a polygon by listing its vertices consecutively. For instance, PQRST and QPTSR are two correct names for the polygon above. Example 1: Identifying Polygons State whether the figure is a polygon. If it is not, explain why. • Not D- because D has a side that isn’t a segment – it’s an arc. • Not E- because two of the sides intersect only one other side. • Not F- because some of its sides intersect more than two sides. A C B F E D Figures A, B, and C are polygons. Polygons are named by the number of sides they have – MEMORIZE Number of Sides Type of Polygon 3 Triangle 4 Quadrilateral 5 Pentagon 6 Hexagon 7 Heptagon Polygons are named by the number of sides they have – MEMORIZE Number of sides Type of Polygon 8 Octagon 9 Nonagon 10 Decagon 12 Dodecagon n n-gon Convex or concave? • Convex if no line that contains a side of the polygon contains a point in the interior of the polygon. • Concave or non-convex if a line does contain a side of the polygon containing a point on the interior of the polygon. See how this crosses a point on the inside? Concave. See how it doesn’t go on the Inside-- convex Convex or concave? • Identify the polygon and state whether it is convex or concave. A polygon is EQUILATERAL If all of its sides are congruent. A polygon is EQUIANGULAR if all of its interior angles are congruent. A polygon is REGULAR if it is equilateral and equiangular. P Ex. : Interior Angles of a Quadrilateral 80° 70° x° x°+ 2x° + 70° + 80° = 360° 3x + 150 = 360 3x = 210 x = 70 Q 2x° R • Sum of the measures of int. s of • A quadrilateral is 360° • Combine like terms • Subtract 150 from each side. • Divide each side by 3. Find m Q and mR. mQ = x° = 70° mR = 2x°= 140° ►So, mQ = 70° and mR = 140° S Investigation Activity • Sketch polygons with 4, 5, 6, 7, and 8 sides • Divide Each Polygon into triangles by drawing all diagonals that are possible from one vertex • Multiply the number of triangles by 180 to find the sum of the measures of the angles of each polygon. 1) Look for a pattern. Describe any that you have found. 2) Write a rule for the sum of the measures of the angles of an n-gon Polygon Angle-Sum Theorem • The sum of the measures of the angles of an n-gon is (n-2)180 • Ex: Find the sum of the measures of the angles of a 15-gon • Sum = (n-2)180 • = (15-2)180 • = (13)180 = 2340 Example • The sum of the interior angles of a polygon is 9180. How many sides does the polygon have? • • • • • Sum = (n-2)180 9180 = (n-2)180 51 = n-2 53 = n The polygon has 53 sides. Polygon Exterior Angle-Sum Theorem • The sum of the measures of the exterior angles of a polygon, one at each vertex, is 360. • An equilateral polygon has all sides congruent • An equiangular polygon has all angles congruent • A regular polygon is both equilateral and equiangular.