6-1 Poly-Sum angle Theorems

Question 1

EX 1: Explore Polygon Interior Angle Sums

How do you calculate the sum of all interior angle measures in a polygon?

Answer 1

180(number of sides figure has - 2 )

3-sided 180(1)
4-sided 180(2)
5-sided 180(3)
6-sided 180(4)
7-sided 180(5)
8-sided 180(6)
9-sided180(7)
10-sided 180(8)
n-sided 180(n - 2)

Question 2

What is a regular polygon?

Answer 2

A polygon that has equal side lengths.

For 3-sided figures, equilateral triangle.
For 4-sided figures, a square.
For 5-sided figures, a regular pentagon
For 6-sided figures, a regular hexagon.
For 7-sided figures, a regular heptagon.
For 8-sided figures, a regular octagon.
For 9-sided figures, a regular nonagon.
For 10-sided figure, a regular decagon.

Question 3

What is the name of a 4-sided figure?

Answer 3

quadrilateral

a 4-sided figure with equal side lengths is called a square

Question 4

What is the name of a 5-sided figure?

Answer 4

pentagon

a 5-sided figure with equal sides is called a regular pentagon

Question 5

What is the name of a 6-sided figure?

Answer 5

hexagon

a 6-sided figure with equal side lengths is called a regular hexagon

Question 6

What is the name of a 7-sided figure?

Answer 6

heptagon

a 7-sided figure with equal side lengths is called a regular heptagon

Question 7

What is the name of a 8-sided figure?

Answer 7

octagon
an 8-sided figure with equal side lengths is called a regular octagon

Question 8

What is the name of a 9-sided figure?

Answer 8

nonagon
a 9-sided figure with equal side lengths is called a regular nonagon

Question 9

What is the name of a 10-sided figure?

Answer 9

decagon
a 10-sided figure with equal side lengths is called a regular decagon

Question 10

EX 1: Explore Polygon Interior Angle Sums

How do you calculate the measure of each individual angles in a regular n-gon?

Answer 10

(180(number of sides - 2))
divided by number of sides

Question 11

EX 3: Understand Exterior Angle Measures of a Polygon

What is the sum of the exterior angles of a polygon

Answer 11

always 180 degrees

no matter how many sides the figure has

Question 12

EX 4: Find an Exterior Angle Measure

How do you calculate the measures of exterior angles that are represented by numbers and/or algebraic expressions?

Answer 12

Write a statement to include sum of every exterior angle = 180

Fill in information given for each angle.

Solve for the value of the variable.

Substitute the value of the variable and simplify to determine the measures of the angles.

Question 13

EX 5: Find the Measures of Interior Angles

How do you calculate the interior angle measures of a figure when the side lengths given are variable or algebraic terms?

Answer 13

Create an equation.

sum of all angle measures = 180(n-2)

Solve for the value of the variable.

Substitute the value of the variable into the expression for any angle to calculate the angle measure for that angle.

Question 14

sum of all angles in a triangle

Answer 14

180 degrees

Question 15

measure of each angle in an equilateral triangle

Answer 15

180 degrees /3 angles

= 60 degrees

Question 16

sum of all angles in a quadrilateral

Answer 16

180(number of sides - 2)
= 180(4 - 2)
= 180(2)
= 360

Question 17

measure of one angle in a square

(regular quadrilateral)

Answer 17

180(number of sides - 2)
= 180(4 - 2)
= 180(2)
= 360 degrees

360/4 = 90 degrees

Question 18

sum of all interior angles in a pentagon

(5-sided figure)

Answer 18

180(number of sides - 2)
= 180(5 - 2)
= 180(3)
= 540 degrees

Question 19

measure of one angle in a regular pentagon

(5-sided figure with all side of equal length)

Answer 19

180(number of sides - 2)
= 180(5 - 2)
= 180(3)
= 540 degrees

540/5 sides = 108 degrees

Question 20

sum of all interior angle measures in a hexagon

(6-sided figure)

Answer 20

180(number of sides - 2)
= 180(6 - 2)
= 180(4)
= 720 degrees

Question 21

measure of one angle in a regular hexagon

(6-sided figure with sides of equal length)

Answer 21

180(number of sides - 2)
= 180(6 - 2)
= 180(4)
= 720 degrees

720/6 sides = 120 degrees

Question 22

sum of all interior angles in an heptagon

(7-sided figure)

Answer 22

180(number of sides - 2)
= 180(7 - 2)
= 180(5)
= 900 degrees

Question 23

measure of one angle in a regular heptagon

(7-sided figure with sides of equal length)

Answer 23

180(number of sides - 2)
= 180(7 - 2)
= 180(5)
= 900 degrees

900/5 sides = 180 degrees

Question 24

measure of one angle in a regular octagon

(8-sided figure with sides of equal length)

Answer 24

180(number of sides - 2)
= 180(8 - 2)
= 180(6)
= 1080 degrees

1080/8 sides = 135 degrees

Question 25

sum of all interior angles in an octagon

(8-sided figure)

Answer 25

180(number of sides - 2)
= 180(8 - 2)
= 180(6)
= 1080 degrees