Sum of Interior and Exterior Angles in Polygons

Essential Question – How can I find angle measures in polygons without using a

protractor?

Key Standard – MM1G3a

Warm-Up

(Triangles)

Polygons• A polygon is a closed figure formed by a finite number

of segments such that:1. the sides that have a common endpoint

are noncollinear, and2. each side intersects exactly two other

sides, but only at their endpoints.

Nonexamples

Polygons• Can be concave or convex.

Concave Convex

Polygons are named by number of sidesNumber of Sides Polygon

34

5

6

78

9

10

12

n

TriangleQuadrilateral

Pentagon

Hexagon

Heptagon

Octagon

Nonagon

Decagon

Dodecagon

n-gon

Regular Polygon

• A convex polygon in which all the sides are congruent and all the angles are congruent is called a regular polygon.

• Draw a: Quadrilateral Pentagon Hexagon Heptagon

Octogon • Then draw diagonals to create triangles.

– A diagonal is a segment connecting two nonadjacent vertices (don’t let segments cross)

• Add up the angles in all of the triangles in the figure to determine the sum of the angles in the polygon.

• Complete this table

Polygon # of sides # of triangles Sum of interior angles

Polygon # of sides # of triangles Sum of interior angles

Triangle

Quadrilateral

Pentagon

Hexagon

Heptagon

Octagon

n-gon

3

4

5

6

7

8

n

3

4

5

6

n - 2

2

1 180°

2 · 180 = 360°

3 · 180 = 540°

4 · 180 = 720°

5 · 180 = 900°

6 · 180 = 1080°

(n – 2) · 180°

Polygon Interior Angles TheoremThe sum of the measures of the interior angles

of a convex n-gon is (n – 2) • 180.Examples – 1. Find the sum of the measures of the interior angles of a 16–

gon.2. If the sum of the measures of the interior angles of a convex

polygon is 3600°, how many sides does the polygon have.

3. Solve for x.

4x - 2

82

108

2x + 10

180n – 360 = 3600 + 360 + 360

108 + 82 + 4x – 2 + 2x + 10 = 360

6x + 198 = 360

6x = 162 6 6

Draw a quadrilateral and extend the sides.There are two sets of angles formed when the sides of a polygon are extended. • The original angles are called interior angles. • The angles that are adjacent to the interior angles are called exterior angles.

These exterior angles can be formed when any side is extended. What do you notice about the interior angle and the exterior angle?

What is the measure of a line?

What is the sum of an interior angle with the exterior angle?

They form a line.

180°

180°

If you started at Point A, and followed along the sides of the quadrilateral making the exterior turns that are marked, what would happen?You end up back where you started or you would make a circle.

What is the measure of the degrees in a circle?

A

BC

D

360°

• The sum of the measures of the exterior angles of a convex polygon, one at each vertex, is 360°.

• Each exterior angle of a regular polygon is 360 n

where n is the number of sides in the polygon

Polygon Exterior Angles Theorem

54⁰

68⁰

65⁰

(3x + 13)⁰

60⁰

(4x – 12)⁰

Find the value for x. Sum of exterior angles is 360°

(4x – 12) + 60+ (3x + 13) + 65 + 54+ 68 = 360

7x + 248 = 360

– 248 – 248

7x = 112

7 7

Example

What is the sum of the exterior angles in an octagon?

What is the measure of each exterior angle in a regular octagon?

360°

360°/8

= 45°

TriangleTriangleInequalitInequalit

yy(Triangle (Triangle Inequality Inequality Theorem)Theorem)

Objectives:Objectives:

– recall the primary parts of a triangle– show that in any triangle, the sum of the lengths of

any two sides is greater than the length of the third side

– solve for the length of an unknown side of a triangle given the lengths of the other two sides.

– solve for the range of the possible length of an unknown side of a triangle given the lengths of the other two sides

– determine whether the following triples are possible lengths of the sides of a triangle

Triangle Inequality TheoremTriangle Inequality Theorem

• The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

AB + BC > ACAB + AC > BCAC + BC > AB AA

BB

CC

a. 3 ft, 6 ft and 9 ft• 3 + 6 > 93 + 6 > 9b. 5 cm, 7 cm and 10 cm • 5 + 7 > 105 + 7 > 10• 7 + 10 > 57 + 10 > 5• 5 + 10 > 75 + 10 > 7

c. 4 in, 4 in and 4 in• Equilateral: 4 + 4 > 4Equilateral: 4 + 4 > 4

Is it possible for a triangle to have sides with the given lengths?

Explain.

(YES)(YES)

(NO)(NO)

(YES)(YES)

a. 6 ft and 9 ft• 9 + 6 > x, x < 15• x + 6 > 9, x > 3• x + 9 > 6, x > – 3 • 15 > x > 3

b. 5 cm and 10 cm

c. 14 in and 4 in

Solve for the length of an unknown side (XX) of a triangle given the

lengths of the other two sides.

5 < x < 15

10< x < 18

Solve for the range of the possible value/s of x, if the triples represent the lengths of the three sides of a

triangle.

• Examples:a. x, x + 3 and 2xb. 3x – 7, 4x and 5x – 6 c. x + 4, 2x – 3 and 3xd. 2x + 5, 4x – 7 and 3x + 1

TRIANGLE TRIANGLE INEQUALITYINEQUALITY

(ASIT and SAIT)(ASIT and SAIT)

OBJECTIVES:

recall the Triangle Inequality Theorem state and identify the inequalities relating sides and angles differentiate ASIT (Angle – Side Inequality Theorem) from SAIT

(Side – Angle Inequality Theorem) and vice-versa identify the longest and the shortest sides of a triangle given

the measures of its interior angles identify the largest and smallest angle measures of a triangle

given the lengths of its sides

INEQUALITIES RELATING SIDES AND ANGLES:

ANGLE-SIDE INEQUALITY THEOREM: If two sides of a triangle are not congruent,

then the larger angle lies opposite the longer side.If AC > AB, then mB > mC.

SIDE-ANGLE INEQUALITY THEOREM: If two angles of a triangle are not congruent,

then the longer side lies opposite the larger angle.angle.If mB > mC, then AC > AB. A

C

B

I. List the sides of each triangle in ascending order.

EXAMPLES:

U

I

E

b.

46

P

O

N

a.

59

61

M

E

L

c.70

P A

T

d.

79

42

JR

E

e.

31

73

PO, ON, PN

UE, IE, UI

ME & EL, ML

AT, PT, PA

JR, RE, JE

Triangle Inequality

(Exterior Angle Theorem)

Objectives:• recall the parts of a triangle• define exterior angle of a triangle• differentiate an exterior angle of a triangle

from an interior angle of a triangle• state the Exterior Angle theorem (EAT) and

its Corollary• apply EAT in solving exercises• prove statements on exterior angle of a

triangle

Exterior Angle of a Polygon:

• an angle formed by a side of a and an extension of an adjacent side.

• an exterior angle and its adjacent interior angle are linear pair

1 2

3

4

Exterior Angle Theorem:

• The measure of each exterior angle of a triangle is equal to the sum of the measures of its two remote interior angles.

• m1 = m3 + m4 1 2

3

4

Exterior Angle Corollary:

• The measure of an exterior angle of a triangle is greater than the measure of either of its remote interior angles.

• m1 > m3 and m1 > m4

1 2

3

4

Examples: Use the figure on the right to answer nos. 1- 4.

1. The m2 = 34.6 and m4 = 51.3, solve for the m1.

2. The m2 = 26.4 and m1 = 131.1, solve for the m3 and m4.

3. The m1 = 4x – 11, m2 = 2x + 1 and m4 = x + 18. Solve for the value of x, m3, m1 and m2.

4. If the ratio of the measures of 2 and 4 is 2:5 respectively. Solve for the measures of the three interior angles if the m1 = 133.

1

2

3

4