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Section 6.1 Angles of Polygons

Section 6.1 Angles of Polygons. A polygon is a plane figure with the following requirements: 1. formed by 3 or more segments called sides 2. each side intersects exactly two other sides, one at each endpoint. Identify the polygons. D. No. Yes. No. No.

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Section 6.1 Angles of Polygons

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  1. Section 6.1 Angles of Polygons • A polygon is a plane figure with the following requirements: • 1. formed by 3 or more segments called sides • 2. each side intersects exactly two other sides, one at each endpoint.

  2. Identify the polygons D No Yes No No

  3. Polygons are named by the number of sides they have.

  4. Each endpoint of a side is also called a vertex of the polygon. • A polygon is named by listing its vertices consecutively with the name of the shape first. • Ex. Polygon MOPW • Square TRZX or

  5. Name the polygon in 2 different ways One possible name: Hexagon BECAFD Another possible name: Hexagon FACEBD

  6. Convex polygon – no line that contains a side also contains a point in the interior of the polygon Concave polygon – a line that contains a side also contains a point in the interior of the polygon.

  7. Diagonals of polygons • A segment that connects any two non consecutive vertices.

  8. Sum of the interior angles • The sum of the angles in a triangle is 180. • How many triangles are contained in this quadrilateral? • So what is the sum of the angles in a quadrilateral?

  9. How many triangles are in this pentagon? • What is the sum of the angles in a pentagon?

  10. What is the pattern between the number of sides and the number of triangles formed? • What is the pattern between the number of triangles formed and the sum of the interior angles? • Formula?

  11. Formula for interior angles: • Where S = sum of the interior angles and n = number of sides of the polygon.

  12. Find the sum of the interior angles • 1. A hexagon • 2. A 15 – gon • 3.

  13. Find the polygon • How many sides does a polygon have if the sum of its interior angles is…. • 1. 1080? • 2. 900? • 3. 500?

  14. Algebra examples • First, find the sum of the interior angles based on the shape of the figure. • Then add the angles together and set it equal to the sum of the interior angles.

  15. Ex. 1 Find the measure of each interior angle Sum of interior angles: Si = (4 – 2)180 = 360 360 = x + 2x -15 + x + 2x - 15 360 = 6x - 30 390 = 6x x = 65

  16. Ex. 2 Find the measure of each interior angle.

  17. Ex. 3 Find the measure of each interior angle. Sum of interior angles: Si = (6 – 2)180 = 720 720 = 7x + 7x + 4x + 7x + 7x + 4x 720 = 36x x = 20

  18. Ex. 4 Find the measure of each interior angle.

  19. An exterior angle is formed when one side at a vertex of a polygon is extended.

  20. The sum of the exterior angles, one at each vertex, is 360. • Find the sum of the exterior angles of a… • 1. quadrilateral • 2. 100-gon • 3. nonagon

  21. Homework: p. 321 #1, 4, 7-14, 19-22, 33, 34

  22. Day 2

  23. Regular polygons • A convex polygon which all the sides are congruent and all the angles are congruent.

  24. Regular polygon examples • To find the measure of one interior angle of a regular polygon, first find the sum then divide by the number of sides • =

  25. Find the measure of an interior angle of a regular dodecagon. • = • =150

  26. Find the measure of an interior angles of a regular 30-gon.

  27. The measure of an interior angle of a regular polygon is 135. Find the number of sides. • Substitute: • 135n = (n – 2)180 • 135n = 180n – 360 • n = 8

  28. The measure of an interior angle of a regular polygon is 171. Find the number of sides.

  29. Find the measure of an interior and an exterior angle of a regular decagon.

  30. HW: p. 321 2,3, 5, 6, 15-18, 23-29, 39, 40

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