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Angle Measures in Polygons - Finding Sum, Exterior, and Interior Angles

Learn how to find the sum of interior angles, measure of exterior angles, and determine the number of sides in a polygon using angle measures. Includes examples and word problems.

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Angle Measures in Polygons - Finding Sum, Exterior, and Interior Angles

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  1. Find Angle Measures in Polygons Ch 5.4

  2. Vocab • Consecutive vertices: vertices that are next to each other or part of the same side.

  3. Diagonal • A diagonal is a segment that connects 2 non-consecutive edges.

  4. Polygon Interior Angles Theorem • Theorem

  5. What is the sum of the interior angles of… • A triangle: • A quadrilateral: • A pentagon: • A hexagon: • A heptagon:

  6. Corollary to Interior Angle Theorem

  7. If I have the Sum of the interior angles how do I figure out how many sides there are?

  8. If I have the Sum of the interior angles how do I figure out how many sides there are?

  9. How would you find x? What kind of figure is this? What is the sum of the interior angles?

  10. What kind of figure is this? What is the sum of the interior angles of a pentagon? (5 – 2)(180) = 540 110+92+100+84 + x = 540 386 + x = 540 x = 154

  11. What kind of figure is it?How do you find x? What is the sum of the interior angles of a heptagon? (7 – 2)(180) = 900 145+112+99+133+156+2x+x = 900 645 + 3x = 900 3x = 255 x = 85

  12. Finding exterior angles

  13. How do you find x? 67 + 96 + 59 + 86 + x = 360 308 + x = 360 x = 52

  14. Find x X = 71

  15. Word Problems • What is the measure of one exterior angle in a regular triangle? • Recall that the sum of any convex polygon’s exterior angles is 360. • Then divide by how many exterior angles there are in the shape. • 360 ÷ 3 = 120 120 ̊

  16. What is the measure of an exterior angle in a regular 10-gon? • 360 ÷ 10 = 36 ̊

  17. The measures of the exterior angles of a convex pentagon are 54, 72, 2x, 3x and x. What is the measure of the largest angle? • The sum of the measure of the exterior angles equals 360, so we’ll add the angles and find x. • 54 + 72 + 2x + 3x + x = 360 • 126 + 6x = 360 • 6x = 234 • X = 39 • 3x = 3(39) = 117

  18. How do you find how many sides a figure has, when given one angle measure? • How do you find the sum of the interior angles? • (n – 2)180 = interior angle sum • If we have a regular figure all the angles are the same, so we would divide the interior angle sum by the number of sides, n, to find the measure of one angle.

  19. An interior angle of a regular polygon has a measure of 150 ̊. How many sides does the figure have? Multiply both sides by n! Distribute the 180! Subtract the 180n!

  20. An interior angle of a regular polygon has a measure of 120 ̊. How many sides does the figure have? Multiply both sides by n! Distribute the 180! Subtract the 180n!

  21. Find the measure of and interior and exterior angle of a regular hexagon. Find the measure of and interior and exterior angle of a regular 20-gon. A regular polygon has an angle measure of 108 ̊ how many sides does the polygon have?

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