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**3.5 The Polygon Angle-Sum Theorems**

Geometry Mr. Barnes

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**Objectives: To Classify Polygons**

To find the sums of the measures of the interior and exterior angles of polygons.

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Definitions: SIDE 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.

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**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. Figures A, B, and C are polygons.

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**Polygons are named by the number of sides they have – MEMORIZE**

Type of Polygon 3 Triangle 4 Quadrilateral 5 Pentagon 6 Hexagon 7 Heptagon

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**Polygons are named by the number of sides they have – MEMORIZE**

Type of Polygon 8 Octagon 9 Nonagon 10 Decagon 12 Dodecagon n n-gon

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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 it doesn’t go on the Inside-- convex See how this crosses a point on the inside? Concave.

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**Convex or concave? CONCAVE CONVEX**

Identify the polygon and state whether it is convex or concave. 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. CONVEX

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**Ex. : Interior Angles of a Quadrilateral**

80° Ex. : Interior Angles of a Quadrilateral 70° 2x° x° x°+ 2x° + 70° + 80° = 360° 3x = 360 3x = 210 x = 70 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°

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**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. Look for a pattern. Describe any that you have found. Write a rule for the sum of the measures of the angles of an n-gon

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**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

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Example The sum of the interior angles of a polygon is 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.

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**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.

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