The Triangle Sum TheoremThe Triangle Sum Theorem

Theorems, Postulates, & Definitions

The Parallel Postulate: Given a line and a point not on the line, there is one and only one line that contains the given point and is parallel to the given line.

Triangle Sum Theorem: The sum of the measures of the angles of a triangle is 180°.

Exterior Angle Theorem: The measure of an exterior angle of a triangle is equal to the sum of the measures of the remote interior angles.

1

2

3

EXTERIOR ANGLE

REMOTE INTERIOR

ANGLES

4

An exterior angle is formed by one side of a triangle and the extension of another side.

Remote interior angles are those interior angles of a triangle not adjacent to a given exterior angle.

Exterior Angle TheoremExterior Angle Theorem

The measure of an exterior angle of a triangle is equal to the sum of the measures of the

two remote interior angles.

1

2

3

Through a point not on a given line, there is one and only one line that goes through that pointthat is parallel to the given line.

The Parallel Postulate:The Parallel Postulate:

The three angles in a triangle add up to be 180º.

The Triangle Sum TheoremThe Triangle Sum Theorem

x + y + z = 180°

x°

y°

z°

Triangle Sum Theorem ProofTriangle Sum Theorem ProofGIVEN: ∆ABC and BD || ACPROVE: m 1 + ∠ m 2 + ∠ m 3 = 180∠

Statements Reasons

BDAC

4 + 2 + 5 =180

1 4 and 3 5

1 + 2 + 3 = 180

180 in a straight line

Given

Alternate Interior ’s are

Substitution Property

Find the value of the variable.Find the value of the variable.

a.a. b.b. c.c.x = 25x = 25 x = 9x = 9 x = 47x = 47

d.d. e.e. f.f.x = 36x = 36 x = 25x = 25 x = 45x = 45

g.g. h.h. i.i.x = 15x = 15 x = 134x = 134 x = 75x = 75

Find the value of the variable(s).Find the value of the variable(s).

a.a. x = 17.5x = 17.5

c.c. x = 53x = 53 d.d. x = 22x = 22

e.e. x = 32x = 32 f.f. x = 30x = 30

b.b. x = 107 y = 87x = 107 y = 87

x

Find the value of the variable(s).Find the value of the variable(s).

a.a. b.b. c.c.x = 35 y = 37x = 35 y = 37 x = 118 y = 96x = 118 y = 96 x = 85 y = 65x = 85 y = 65

d.d. e.e. f.f.x = 26 y = 64x = 26 y = 64 x = 43 y = 32x = 43 y = 32 x = 62 y = 28x = 62 y = 28

a. ma. m1 1 b. mb. m22 c. mc. m33= 50= 50 = 130 = 130 = 50 = 50

d. md. m44 e. me. m55 f. mf. m66= 130 = 130 = 40 = 40 = 30 = 30

Find the measure of each numbered angle in the figure.Find the measure of each numbered angle in the figure.

Find the measure of each numbered angle in the figure.Find the measure of each numbered angle in the figure.

a. ma. m1 1 b. mb. m22 c. mc. m33= 70 = 70 = 110 = 110

d. md. m44 e. me. m55= 102 = 102 = 37 = 37

54 3

21

41 64 29

3238

Assignment

3.5A and 3.5B3.5A and 3.5BSection 8 - 25Section 8 - 25

Angles in PolygonsAngles in Polygons

Objective

• Develop and use formulas for the sums of the measures of interior and exterior angles of a polygon.

Sum of interior angles in a polygonSum of interior angles in a polygon

We already know that the sum of the interior angles in any triangle is 180°.

a + b + c = 180 °

Do you know the sum of the interior angles for any other polygons?

a b

c

We also know that the sum of the interior angles in any quadrilateral is 360°.

a

bc

d

a + b + c + d = 360 °

Sum of the interior angles in a polygon

A quadrilateral can be divided into two triangles …

… and a pentagon can be divided into three triangles.

How many triangles can a hexagon be divided into?

A hexagon can be divided into four triangles.

The number of triangles that a polygon can be divided into is always two less than the number of sides.

The number of triangles that a polygon can be divided into is always two less than the number of sides.

We can say that:

A polygon with s sides can be divided into (s – 2) triangles.

The sum of the interior angles in a triangle is 180°.

So:

The sum of the interior angles in an s-sided polygon is (s – 2) × 180°.The sum of the interior angles in an s-sided polygon is (s – 2) × 180°.

Sum of the interior angles in a polygon

Interior angles in regular polygonsInterior angles in regular polygons

A regular polygon has equal sides and equal angles.

We can work out the size of the interior angles in a regular polygon as follows:

Name of regular polygon

Sum of the interior angles

Size of each interior angle

Equilateral triangle180° 180° ÷ 3 = 60°

Square2 × 180° = 360° 360° ÷ 4 = 90°

Regular pentagon3 × 180° = 540° 540° ÷ 5 = 108°

Regular hexagon4 × 180° = 720° 720° ÷ 6 = 120°

Interior and exterior angles in an Interior and exterior angles in an equilateral triangleequilateral triangle

In an equilateral triangle,

60°

60°

Every interior angle measures 60°.

Every exterior angle measures 120°.

120°

120°

60°120°

The sum of the interior angles is 3 × 60° = 180°.

The sum of the exterior angles is 3 × 120° = 360°.

Interior and exterior angles in a square

In a square,

Every interior angle measures 90°.

Every exterior angle measures 90°.

The sum of the interior angles is 4 × 90° = 360°.

The sum of the exterior angles is 4 × 90° = 360°.

90° 90°

90° 90°

90°

90°

90°

90°

Interior and exterior angles in a regular pentagon

In a regular pentagon,

Every interior angle measures 108°.

Every exterior angle measures 72°.

The sum of the interior angles is 5 × 108° = 540°.

The sum of the exterior angles is 5 × 72° = 360°.

108°

108° 108°

108° 108°

72°72°

72°

72°

72°

Interior and exterior angles in a regular Interior and exterior angles in a regular hexagonhexagon

In a regular hexagon,

Every interior angle measures 120°.

Every exterior angle measures 60°.

The sum of the interior angles is 6 × 120° = 720°.

The sum of the exterior angles is 6 × 60° = 360°.

120° 120°

120° 120°

120° 120°

60°

60°

60°

60°

60°

60°

The sum of exterior angles in a polygonThe sum of exterior angles in a polygon

For any polygon, the sum of the exterior angles is 360°.

The sum of the interior angles is (s – 2) × 180°.

• A Polygon can either be convex or concave.

• If a polygon is convex then no sides go through the interior of the polygon.

• (All vertices point outside the polygon.)

• If a polygon is concave then it is not convex. A side goes through the interior of the polygon.

• (At least one vertex points inside the polygon.)

Label the polygons as convex or concave?

• convex • concave • concave • convex

Theorems, Postulates, & Definitions

Sum of the Interior Angles of a Polygon: The sum, a, of the measures of the interior angles of a polygon with s sides is given by

a = (s – 2)180.

Theorems, Postulates, & Definitions

Sum of the Exterior Angles of a Polygon: The sum of the measures of the exterior angles of a polygon is 360.

Complete the chart below:Complete the chart below:Regular polygon

Sides Sum of the

Interior Angles

Measure of One Interior

Angle

Measure of One Exterior

Angle

Triangle 3

Quadrilateral 4

Pentagon 5

Hexagon 6

Heptagon 7

Octagon 8

Nonagon 9

n-gon n

180 60 120

360 90 90540 108 72

720 120 60900

3751 128 4

7

1080 135 45

1260 140 40(n - 2)180

n n360

(n -2 )180

An exterior angle measure of a regular polygon is given. Find the number of its sides and the measure of each interior angle.

a. 120°

c. 36°

b. 72°

d. 24°

Example: The exterior angle measure of a regular polygon is 45. Find the number of sides.

Since there is 360 in the exterior of a figure with any number of sides, divide 360 by 45 to find the number of sides.

sides845

360

3 sides 5 sides

10 sides 15 sides

Name the convex polygon whose interior angle measures have each given sum.

a. 540° c. 1800°b. 900° d. 2520°

Example: The sum of the interior angle measures of a regular polygon is 720. Find the number of sides.Use the formula for the sum of the interior angles of a polygon. AnglesInteriortheofSum180)2s(

5 sides Pentagon

7 sides Heptagon

12 sidesDodecagon

16 sides 16-gon

720180)2s(

180

720)2s(

42s hexagon6s

Find the value of each variableFind the value of each variable

a.a. b.b. c.c.a = 18a = 18 r = 15r = 15 y = 22.5y = 22.5 d.d.

e.e. f.f.

n = 24n = 24

x = 27x = 27 s = 18s = 18 g.g. h.h.

i.i.

x = 61.5x = 61.5 x = 72x = 72

x = 124x = 124 j.j. n = 24n = 24 k.k. a = 30a = 30 l.l. x = 90 y = 75x = 90 y = 75z = 120z = 120

AssignmentAssignment

3.6A and 3.6B3.6A and 3.6BSection 8 - 37Section 8 - 37