6-1 Angles of Polygons6-1 Angles of Polygons

You named and classified polygons.

• Find and use the sum of the measures of the interior angles of a polygon.

• Find and use the sum of the measures of the exterior angles of a polygon.

DefinitionsDefinitions

A A polygonpolygon is a plane figure whose sides is a plane figure whose sides are three or more coplanar segments that are three or more coplanar segments that intersect only at their endpoints (the intersect only at their endpoints (the vertices). Consecutive sides cannot be vertices). Consecutive sides cannot be collinear, and no more than two sides can collinear, and no more than two sides can meet at any one vertex.meet at any one vertex.

DefinitionDefinition

A A diagonal of a polygondiagonal of a polygon is a line is a line segment whose endpoints are any two segment whose endpoints are any two nonconsecutive vertices of the polygon.nonconsecutive vertices of the polygon.

diagonal

Classification of PolygonsClassification of Polygons

3 sides Triangle

4 sides Quadrilateral

5 sides Pentagon

6 sides Hexagon

7 sides Heptagon

8 sides Octagon

9 sides Nonagon

10 sides Decagon

12 sides Dodecagon

20 sides Icosagon

Polygon interior anglesPolygon interior angles

What happens to the sum of the degrees of What happens to the sum of the degrees of the inside angles as the number of sides the inside angles as the number of sides increases?increases?

180° 360° 540°720°

Is there an easier way to figure out the sum of the degrees of the inside angles for any polygon?

Angle-Sum Theorem for PolygonsAngle-Sum Theorem for Polygons

The sum of the measures of the interior The sum of the measures of the interior angles of a convex polygon with n sides is angles of a convex polygon with n sides is given by given by S = (n S = (n − 2)180°− 2)180°

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A. Find the sum of the measures of the interior angles of a convex nonagon.

A nonagon has nine sides. Use the Polygon Interior Angles Sum Theorem to find the sum of its interior angle measures.

(n – 2) ● 180= (9 – 2) ● 180 n = 9

= 7 ● 180 or 1260 Simplify.

Answer: The sum of the measures is 1260.

B. Find the measure of each interior angle of parallelogram RSTU.

Since n=4, the sum of the measures of the interior angles is Write an equation to express the sum of the measures of the interior angles of the polygon.

Step 1 Find x.

Sum of measures of interior angles Substitution Combine like terms.Subtract 8 from each side.

Divide each side by 32.

Step 2 Use the value of x to find the measure of each angle.

Answer: mR = 55, mS = 125, mT = 55, mU = 125

mR = 5x= 5(11) or 55

mS = 11x + 4= 11(11) + 4 or 125

mT = 5x= 5(11) or 55

mU = 11x + 4= 11(11) + 4 or 125

A. 900

B. 1080

C. 1260

D. 1440

A. Find the sum of the measures of the interior angles of a convex octagon.

A. 130°

B. 128.57°

C. 140°

D. 125.5°

A pottery mold makes bowls that are in the shape of a regular heptagon. Find the measure of one of the interior angles of the bowl.

The sum of the measures of the The sum of the measures of the interior angles of a convex polygon interior angles of a convex polygon

is 900is 900°. Find °. Find the number of sides the number of sides of of the polygon.the polygon.

S = (nS = (n− 2)180°− 2)180°

900 = (n − 2)180°900 = (n − 2)180°

900900÷180÷180 = (n − 2)180° = (n − 2)180° ÷180÷180

5 = n − 25 = n − 2

5 5 + 2+ 2 = n − 2 = n − 2 + 2+ 2

7 = n7 = n

The measure of an interior angle of a regular polygon is 150. Find the number of sides in the polygon.Use the Interior Angle Sum Theorem to write an equation to solve for n, the number of sides.

S = 180(n – 2) Interior Angle SumTheorem

(150)n = 180(n – 2) S = 150n

150n = 180n – 360 Distributive Property

0 = 30n – 360 Subtract 150n from eachside.

360 = 30n Add 360 to each side.

12 = n Divide each side by 30.

Answer: The polygon has 12 sides.

A. 12

B. 9

C. 11

D. 10

The measure of an interior angle of a regular polygon is 144. Find the number of sides in the polygon.

Polygon exterior anglesPolygon exterior anglesWhat happens to the sum of the degrees of What happens to the sum of the degrees of

the outside angles as the number of sides the outside angles as the number of sides increases?increases?

360° 360° 360°360°

Is there an easier way to figure out the sum of the degrees of the outside angles for any polygon?

Exterior Angle Theorem for Exterior Angle Theorem for PolygonsPolygons

The sum of the measures of the exterior The sum of the measures of the exterior angles of a convex polygon (one at each angles of a convex polygon (one at each

vertex) is vertex) is 360360°°

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Find the sum of the interior angles Find the sum of the interior angles ANDAND exterior angles for the polygon exterior angles for the polygon

77 sides sides Heptagon Heptagon

Interior angle sum = (n Interior angle sum = (n − 2)180°− 2)180°

= = ((77 − 2)180°− 2)180°

= = (5(5)180°)180°

= 900°= 900°

Exterior angle sum = 360°Exterior angle sum = 360°

Find the sum of the interior angles Find the sum of the interior angles AND AND exterior angles for the polygonexterior angles for the polygon

2222-gon-gon

Interior angle sum = (n Interior angle sum = (n − 2)180°− 2)180°

= = ((2222 − 2)180°− 2)180°

= = (20(20)180°)180°

= 3600°= 3600°

Exterior angle sum = 360°Exterior angle sum = 360°

A. Find the value of x in the diagram.Use the Polygon Exterior Angles Sum Theorem to write an equation. Then solve for x.

Answer: x = 12

5x + (4x – 6) + (5x – 5) + (4x + 3) + (6x – 12) + (2x + 3) +(5x + 5) = 360

(5x + 4x + 5x + 4x + 6x + 2x + 5x) + [(–6) + (–5) + 3 + (–12) + 3 + 5] = 360

31x – 12 = 360

31x = 372

x = 12

A. 72

B. 60

C. 45

D. 90

B. Find the measure of each exterior angle of a regular pentagon.

How do you find the sum of the measures of the How do you find the sum of the measures of the interior angles of a convex polygon?interior angles of a convex polygon?

S = (n −2)180°How do you find the measure of one of the interior

angles of a convex polygon?S = (n −2)180°/n

What is the sum of the measure of the exterior What is the sum of the measure of the exterior angles of a convex polygon?angles of a convex polygon?

360° How do you find the measure of one of the exterior

angles of a convex polygon? 360°/n

6-1 Assignment6-1 Assignment

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