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Objectives

Objectives. Define polygon, concave / convex polygon, and regular polygon Find the sum of the measures of interior angles of a polygon Find the sum of the measures of exterior angles of a polygon. Definition of polygon.

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Objectives

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  1. Objectives • Define polygon, concave / convex polygon, and regular polygon • Find the sum of the measures of interior angles of a polygon • Find the sum of the measures of exterior angles of a polygon

  2. Definition of polygon • A polygon is a closed plane figure formed by 3 or more sides that are line segments; • the segments only intersect at endpoints • no adjacent sides are collinear • Polygons are named using letters of consecutive vertices

  3. Concave and Convex Polygons • A convex polygon has no diagonal with points outside the polygon • A concave polygon has at least one diagonal with points outside the polygon

  4. Regular Polygon Definition • An equilateral polygon has all sides congruent • An equiangular polygon has all angles congruent • A regular polygon is both equilateral and equiangular Note: A regular polygon is always convex

  5. Sum of Interior Angles in Polygons

  6. Example: Sum of Interior Angles The marks in the illustration indicate that m∠X = m∠Y. So the sum of all four interior angles is m∠X + m∠X + 100 + 90 = 360 2 m∠X + 190 = 360 2 m∠X = 170 m∠X = 85 Find m∠ X Solution: The sum of the measures of the interior angles for a quadrilateral is (4 – 2) * 180 = 360

  7. Polygon Exterior Angle Sum Theorem • The sum of the measures of the exterior angles of a polygon, one at each vertex is 360. m∠1 + m∠2 + m∠3 + m∠4 + m∠5 = 360

  8. Example: Exterior Angle Sum What is the measure of an interior angle of a regular octagon? Solution: 8 * exterior angle = 360 (Ext. Angle Sum) exterior angle = 45 interior angle = 180 – exterior angle interior angle = 180 – 45 = 135

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