Polygon Angles: Sum of Interior and Exterior Angles, Lecture notes of Analytical Geometry and Calculus

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8.1 ANGLES OF

POLYGONS

Find and use the sum of the measures of the interior and exterior angles of a polygon

Polygon Interior Angles Sum

A DIAGONAL of a polygon is a segment that connects any two nonconsecutive vertices Triangle 180° Quadrilateral 2 180° = 360° Pentagon 3 180° = 540° Hexagon 4 180° = 720°

Polygons

Sides Name Measure of Interior Angles 3 Triangle 1 ∙ 180° = 180° 4 Quadrilateral 2 ∙ 180° = 360° 5 Pentagon 3 ∙ 180° = 540° 6 Hexagon 4 ∙ 180° = 720° 7 8 9 10 11 12 N Heptagon Octagon Nonagon Decagon Hendecagon Dodecagon 5 ∙ 180° = 900° 6 ∙ 180° = 1080° 7 ∙ 180° = 1260° 8 ∙ 180° = 1440° 9 ∙ 180° = 1620° 10 ∙ 180° = 1800° 𝑛 − 𝑔𝑜𝑛 𝑛 − 2 ∙ 180°

Polygon Interior Angle Theorem

The sum of the interior angle measures of an 𝑛 −sided convex polygon 𝑛 − 2 ∙ 180.

Example 2: Find the measure of each

interior angle of the pentagon 𝐻𝐽𝐾𝐿𝑀 shown 5 sides total 5 − 2 ∙ 180° The sum of the interior angles is 540°

Recall:

A REGULAR POLYGON is a polygon in which all of the sides are congruent and all the angles are congruent

Example 4: The measure of each interior angle of a

polygon is 150. Find the number of sides in the polygon Let 𝑛 be the number of sides. Since all angles of a regular polygon are congruent, the sum of the interior angles can be expressed as 150𝑛 150𝑛 = 𝑛 − 2 ∙ 180 150𝑛 = 180𝑛 − 360 −30𝑛 = − 360 𝑛 = 12 There are 12 sides.

Polygon Exterior Angle Sum

◦ The sum of the exterior angle measures of a convex polygon, one angle at each vertex is 360.

Example 6: Find the measure of each

exterior angle of a regular dodecagon

12 sides Let 𝑛 represent the measure of each exterior angle 12𝑛 = 360 𝑛 = 30 Each exterior angle of a regular dodecagon is 30°.