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Finding the Sum of the Interior Angles of a Convex Polygon

Finding the Sum of the Interior Angles of a Convex Polygon. Fun with Angles Mrs. Ribeiro’s Math Class. Review Terms. Polygons Convex and Concave Polygons Vertex (pl. Vertices). Polygons. A plane shape (two-dimensional) with straight sides.

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Finding the Sum of the Interior Angles of a Convex Polygon

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  1. Finding the Sum of the Interior Angles of a Convex Polygon Fun with Angles Mrs. Ribeiro’s Math Class

  2. Review Terms • Polygons • Convex and Concave Polygons • Vertex (pl. Vertices)

  3. Polygons A plane shape (two-dimensional) with straight sides. Examples: triangles, rectangles and pentagons.Note: a circle is not a polygon because it has a curved side

  4. Types of Polygons

  5. Convex Polygon A convex polygon has no angles pointing inwards. More precisely, no internal angles can be more than 180°.

  6. Concave Polygon If there are any internal angles greater than 180° then it is concave. (Think: concave has a "cave" in it)

  7. Review Terms • Side • Adjacent v. Opposite • Diagonals

  8. Review Concepts • What is the sum of the interior angles of a triangle? • How can we use this to find missing angles in a triangle? a + b + c = 180º

  9. Triangle Sum Theorem What is the measure of the third angle? a + b + c = 180º

  10. Triangle Sum Theorem The measure of the third angle is: The interior angles of a triangle add to 180° The sum of the given angles = 29° + 105° = 134° Therefore the third angle = 180° - 134° = 46°

  11. Divide a Polygon into Triangles • Choose a vertex • Draw a diagonal to the closest vertex at left that is not adjacent • Repeat for additional diagonals until you reach the adjacent at right

  12. Polygons into Triangles Hexagon: Quadrilateral:

  13. Polygons into Triangles • Let’s count triangles!… • Hexagon: Quadrilateral

  14. Rule for Convex Polygons Sum of Internal Angles = (n-2) × 180° Measure of any Angle in Regular Polygon = (n-2) × 180° / n

  15. Example: A Regular Decagon Sum of Internal Angles = (n-2) × 180° (10-2)×180° = 8×180° = 1440° Each internal angle (regular polygon) = 1440°/10 = 144°

  16. Find an interior angle What is the fourth interior angle of this quadrilateral? A 134° B 129° C 124° D 114° Use pencil and paper – work with a shoulder partner

  17. Find an interior angle Sum of interior angles of a quadrilateral: 360° Given angles sum = 113° + 51° + 82° = 246° Fourth angle a + b + c + d = 360º a + b + c = 246º d = 360 º - 246º = 114 º

  18. Working “Backwards” Each of the interior angles of a regular polygon is 156°. How many sides does this polygon have? A 15 B 16 C 17 D 18

  19. Working “Backwards” Use the formula for one angle of a regular n-sided polygon. We know one angle = 156° Now we solve for "n": Multiply both sides by n ⇒ (n - 2) × 180 = 156n Expand (n-2) ⇒ 180n - 360 = 156n Subtract 156n from both sides: ⇒ 180n - 360 - 156n = 0 Add 360 to both sides: ⇒ 180n - 156n = 360 Subtract 180n-156n ⇒ 24n = 360 Divide by 24 ⇒ n = 360 ÷ 24 = 15

  20. References • Johnson, Lauren. (27 April 2006). “Polygons and their interior angles.” University of Georgia. Retrieved (04 Dec. 2011) from http://intermath.coe.uga.edu/tweb/gcsu-geo-spr06/ljohnson/geolp2.doc. • Kuta Software LLC. (2011). “Introduction to Polygons” Infinite Geometry. Retrieved (04 Dec. 2011) from <http://www.kutasoftware.com/FreeWorksheets/GeoWorksheets/6-Introduction%20to%20Polygons.pdf> • Mathopolis.com (2011) “Question 1780 by lesbillgates.” Retrieved (0 Dec. 2011) from <http://www.mathopolis.com/questions/q.php?id=1780&site=1&ref=/geometry/interior-angles-polygons.html&qs=825_826_827_828_1779_829_1780> • Pierce, Rod. (2010). “Interior Angles of Polygons.” MathsisFun.com. Retrieved (04 Dec. 2011) from <http://www.mathsisfun.com/geometry/interior-angles-polygons.html>

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