Quadrilaterals Chapter 6 Angles of Polygons Chapter 6

Quadrilaterals Chapter 6 Angles of Polygons Chapter 6. 1

Types of Polygons # of Sides Type of Polygon 3 Triangle 8 Octagon 4 Quadrilateral 9 Nonagon 5 Pentagon 10 Decagon 6 Hexagon 12 Dodecagon 7 Heptagon n n-gon

Regular Polygons • Equilateral all sides are congruent • Equiangular all angles are congruent • Regular Polygon Equilateral and Equiangular

Identify Polygons Name Quadrilateral Octagon Pentagon Equilateral? Yes Yes Equiangular? No No Yes Regular? No No Yes

Diagonals • A segment that connects two nonconsecutive vertices in a polygon. Diagonals

Interior Angles of a Polygon Angle sum of a Pentagon = 3 (180) = 5400 Angle sum of a Hexagon = 4 (180) = 7200

Interior Angle Sum Theorem • The sum of the measures of the interior angles of any convex n-gon is = (n – 2) • 180 o # of s 180 o per • The measure of each interior angle of a regular n-gon is

Interior Angles of Regular Polygons ARCHITECTURE A mall is designed so that five walkways meet at a food court that is in the shape of a regular pentagon. Find the sum of the measures of the interior angles of the pentagon. Since the pentagon is a convex polygon, we can use the Interior Angle Sum Theorem. Interior Angle Sum Thm. Simplify. Answer: The sum of the measures of the angles is 540.

A decorative window is designed to have the shape of a regular octagon. Find the sum of the measures of the interior angles of the octagon. A. 1440 B. 1260 C. 1080 D. 900

Find the value of X Interior angle sum of a 7 -gon is o (7 -2) • 180 95 o 5 • 180 o 155 o 900 o 128+102+143+95+ 155+137 + X = 900 760 + X = 900 137 o X = 140 X 143 o 102 o 128 o

Co py pro this ble m! The measure of an interior angle of a regular polygon is 135. Find the number of sides in the polygon. Answer: The polygon has 8 sides.

The measure of an interior angle of a regular polygon is 144. Find the number of sides in the polygon. A. 12 B. 9 C. 11 D. 10

Interior Angles of a Quadrilateral (theorem) • The sum of the measures of the interior angles of a quadrilateral is 360 o. 1 2 4 3 m 1 + m 2 + m 3 + m 4 = 360 o

Sample Problem 85 o 2 xo xo 65 o 85 + 65 + x + 2 x = 360 o 150 + 3 x = 360 o 3 x = 210 o x = 70 o

Interior Angles of Nonregular Polygons Find the measure of each interior angle. Since angles is the sum of the measures of the interior

Interior Angles of Nonregular Polygons Sum of measures of int. Substitution Combine like terms. Subtract 8 from each side. Divide each side by 32. Use the value of x to find the measure of each angle. Answer: m R = 55, m S = 125, m T = 55, m U = 125

Find the value of x. A. x = 7. 8 B. x = 22. 2 C. x = 15 D. x = 10

Exterior Angles of a Polygon • The sum of the exterior angles of any convex polygon is 360 o • The measure of each exterior angle of a regular polygon is Sketchpad

Exterior Angles Find the measures of an exterior angle and an interior angle of convex regular nonagon ABCDEFGHJ. At each vertex, extend a side to form one exterior angle.

Exterior Angles The sum of the measures of the exterior angles is 360. A convex regular nonagon has 9 congruent exterior angles. Divide each side by 9. Answer: The measure of each exterior angle is 40. Since each exterior angle and its corresponding interior angle form a linear pair, the measure of the interior angle is 180 – 40 or 140.

Find the measures of an exterior angle and an interior angle of convex regular hexagon ABCDEF. A. 30; 150 B. 45; 135 C. 60; 120 D. 20; 160

Homework Chapter 6. 1 • Pg 321 1 -10, 15 -20, 23 -29