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Triangle Sum Theorem

Triangle Sum Theorem. The sum of the interior measures of the angles of a triangle is 180 degrees. Triangle Exterior Angle Theorem. The measures of each exterior angle of a triangle equals the sum of the measures of its two remote interior angles. Polygon.

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Triangle Sum Theorem

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  1. Triangle Sum Theorem The sum of the interior measures of the angles of a triangle is 180 degrees.

  2. Triangle Exterior Angle Theorem The measures of each exterior angle of a triangle equals the sum of the measures of its two remote interior angles.

  3. Polygon • A closed plane figure formed by 3 or more segments that all lie in one plane

  4. Most Common Polygons Polygons are named by number of sides Triangle 3 4 Quadrilateral Pentagon 5 Hexagon 6 Heptagon 7 8 Octagon 9 Nonagon 10 Decagon 12 Dodecagon n n-gon

  5. An equilateral polygon: All sides congruent. • An equiangular polygon: All angles congruent. • A regular polygon: All the sides and angles congruent. Regular Polygon Equilateral Polygon Equiangular Polygon

  6. Concave • If any part of a diagonal contains points in the exterior of the polygon.

  7. Convex • If no diagonal contains points in the exterior. • A regular polygon is always convex.

  8. 3 1 180° 4 2 2 · 180 = 360° 5 3 3 · 180 = 540° 4 4 · 180 = 720° 6 7 5 5 · 180 = 900° 8 6 6 · 180 = 1080° n n – 2 (n – 2) · 180°

  9. Ex: What is the measure of angle Y in pentagon TODAY?

  10. Polygon Angle-Sum Theorem • The sum of the measures of the interior angles of an n-gon is: Sum = (n – 2)180 • n = the number of sides

  11. Ex: What is the sum of the measures of the interior angles of an octagon? Sum = (n – 2)180 = (8 – 2)180 = 6 * 180 = 1,080°

  12. (n – 2)180 = Sum (n – 2)180 = 3600 180n – 360 = 3600 + 360 + 360 180n = 3960 180 180 n = 22 sides Ex: If the sum of the measures of the interior angles of a convex polygon is 3600°, how many sides does the polygon have.

  13. (n – 2)180 = Sum (n – 2)180 = 2340 180n – 360 = 2340 + 360 + 360 180n = 2,700 180 180 n = 15 sides Ex: If the sum of the measures of the interior angles of a convex polygon is 2340°, how many sides does the polygon have.

  14. Ex: Solve for x Sum = (n – 2)180 4x – 2 108 108 + 82 + 4x – 2 + 2x + 10 = (4 – 2)180 2x + 10 82 6x + 198 = 360 6x = 162 6 6 x = 27

  15. Ex. Find the values of the variables and the measures of the angles. x = 25 1300 900 1150 1150 900

  16. The measure of each interior angle of a regular n-gon is

  17. Ex: What is the measure of each or one interior angle in a regular octagon? (8 – 2)180 / 8 1350

  18. What do you notice about the exterior angles of the polygons below?

  19. Polygon Exterior Angle-Sum Theorem • The sum of the measures of the exterior angles of a polygon, one at each vertex, is 360.

  20. Ex. Find the exterior angle sum of a decagon.

  21. Ex: Find the value of x Sum of exterior angles is 360° (4x – 12) + 60+ (3x + 13) + 65 + 54+ 68 = 360 7x + 248 = 360 – 248 – 248 7x = 112 7 7 x = 12 (4x – 12)⁰ 68⁰ 60⁰ 54⁰ (3x + 13)⁰ 65⁰

  22. Ex: What is the measure of angle 1 in the regular octagon?

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