Triangle Sum Theorem n The sum of the

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Triangle Sum Theorem n The sum of the interior measures of the angles of

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

Triangle Exterior Angle Theorem n The measures of each exterior angle of a triangle

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

Polygon n A closed plane figure formed by 3 or more segments that all

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

Polygons are named by number of sides Number of Sides 3 4 5 6

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

n n n An equilateral polygon: All sides congruent. An equiangular polygon: All angles

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

Concave n If any part of a diagonal contains points in the exterior of

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

Convex n If no diagonal contains points in the exterior. ¨A regular polygon is

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

Polygon # of sides (n) # of triangles Sum of interior angles of a

Polygon # of sides (n) # of triangles Sum of interior angles of a polygon Triangle 3 1 180° Quadrilateral 4 2 2 · 180 = 360° Pentagon 5 3 3 · 180 = 540° Hexagon 6 4 4 · 180 = 720° Heptagon 7 5 5 · 180 = 900° Octagon 8 6 n-gon n n– 2 6 · 180 = 1080° (n – 2) · 180°

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

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

Polygon Angle-Sum Theorem ¨The sum of the measures of the interior angles of an

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

Ex: What is the sum of the measures of the interior angles of an

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°

1. Ex: If the sum of the measures of the interior angles of a

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

1. Ex: If the sum of the measures of the interior angles of a

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

Ex: Solve for x 4 x – 2 82 108 2 x + 10

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

Ex. Find the values of the variables and the measures of the angles. x

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

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

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

Ex: What is the measure of each or one interior angle in a regular

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

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

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

Polygon Exterior Angle-Sum Theorem n The sum of the measures of the exterior angles

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

Ex. Find the exterior angle sum of a decagon.

Ex. Find the exterior angle sum of a decagon.

Ex: Find the value of x Sum of exterior angles is 360° (4 x

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

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

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

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