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Understanding Angles in Polygons: Interior and Exterior Theorems

This section covers the fundamental theorems related to the angles of polygons. The Interior Angles Theorem states that the sum of the interior angles of a convex n-gon is (n-2)180°. For regular polygons, the measure of each interior angle is given by (n-2)180/n. The Exterior Angles Theorem reveals that the sum of the exterior angles of any convex polygon is 360°. It also illustrates how to find properties of regular polygons, such as the type based on exterior angles, and provides examples involving various polygons, including pentagons and nonagons.

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Understanding Angles in Polygons: Interior and Exterior Theorems

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  1. Section 6.2 Angles of Polygons

  2. Theorems • Thm 6.1: Polygon Interior Angles Thm • The sum of the measures of the interior angles of a convex n-gon is (n-2) 180° • Corollary to Thm 6.1 • The measure of each interior angle of a regular n- gon is • (n-2)180 n n for all of these is the number of sides.

  3. Find the measure of each interior angle of a regular pentagon: (n- 2)180° (5-2)180° 3(180)=540° 540/5= 108°

  4. More Thms cont. • Thm 6.2 Polygon Exterior Angles Thm: • The sum of the measures of the exterior angles, one from each vertex, of a convex polygon is 360° • Corollary to Thm 6.2: • The measure of each exterior angle of a regualr n-gon is (360°)/n

  5. Examples: • 1. One exterior angle of a regular convex polygon is 45°. What type of polygon is this? • 45= 360/n • 45n = 360 • n= 8 so octagon • 2. One exterior angle of a regular convex polygon is 30 °. What is the sum of the interior angles? • 30n = 360 • N = 12 • (12- 2)180= 1800°

  6. More exs. • 3. Each interior angle of a regular convex polygon is 60°. How many sides does it have? • 120n= 360 • n= 3 • 4. What is the measure of each interior angle of a regular convex pentagon? • 5n = 360 • N = 72 • 180 – 72= 108 • 5. What is the measure of each interior angle of a regular convex nonagon? • 140° 60

  7. More exs. A polygon has 27 sides, all diagonals are drawn from one vertex, how many diagonals are there? 5 sides 3 triangles 2 diagonals 6 sides 4 triangles 3 diagonals 24 N= sides N -2 = # triangles N- 3 = # of diagonals Do it for an octagon N = 8, triangles=6, diagonals= 5

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