CONFIDENTIAL 1

Geometry

Properties and Attributes of Polygons

CONFIDENTIAL 2

Warm up

Solve by factoring:

1) x2 + 3x – 10 = 0

2) x2 - x – 12 = 0

3) x2 - 12x = - 35

CONFIDENTIAL 3

Today you will learn about the parts of polygon and the ways to classify polygons.

Each segment that forms a polygon is a side of the polygon. The common endpoint of two sides is a vertex of the

polygon. A segment that connects any two nonconsecutive vertices is a diagonal.

Properties and Attributes of Polygons

diagonal

A

E

B

CD

sidevertex

CONFIDENTIAL 4

You can name a polygon by the number of its sides. The table shows the names of some common polygons. Polygon ABCDE in the previous slide is a pentagon.

Number of Slides Name of Polygon

3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

9 Nonagon

10 Decagon

12 Dodecagon

n n - gon

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Identifying Polygon

Tell whether each figure is a polygon. If it is a polygon, name it by the number of its sides:

PolygonPentagon

Not a PolygonPolygonOctagon

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Now you try!

Tell whether each figure is a polygon. If it is a polygon, name it by the number of its sides:

1a 1b 1c

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All the sides are congruent in an equilateral polygon. All the angles are congruent in an equiangular polygon. A regular polygon is one that is both equilateral and equiangular. If a

polygon is not regular, it is called irregular.

A polygon is concave if any part of a diagonal contains points in the exterior of the polygon. If no diagonal contains points in

the exterior, then the polygon is convex.

concave quadrilateral

convex quadrilateral

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Classifying Polygons

Next page ->

Tell whether each polygon is regular or irregular. Tell whether it is concave or convex:

A

irregular convex

CONFIDENTIAL 9

Tell whether each polygon is regular or irregular. Tell whether it is concave or convex:

B

regular convex

C

irregular concave

CONFIDENTIAL 10

Now you try!

Tell whether each polygon is regular or irregular. Tell whether it is concave or convex:

2a 2b

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To find the sum of the interior angles measure of a convex polygon, draw all possible diagonals from one vertex of the

polygon. This creates a set of triangles.

The sum of the angle measures of all the triangles equals the sum measures of the polygon.

Triangle

Quadrilateral

Pentagon Hexagon

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Polygon Number of Slides

Number of Triangles

Sum of Interior Angle Measures

Triangle 3 1 (1) 180° = 180°

Quadrilateral 4 2 (1) 180° = 360°

Pentagon 5 3 (1) 180° = 540°

Hexagon 6 4 (1) 180° = 720°

n - gon n n - 2 (n - 2) 180°

In each convex polygon, the number of triangles formed is two less than the number of sides n. So the sum of the

angle measures of all these triangles is (n - 2) 180°.

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The sum of the interior angle measures of a convex polygon with sides n is

(n - 2) 180°.

Polygon Angle Sum Theorem

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Finding Interior Angle Measures and Sums in Polygons

A) Find the sum of the interior angle measures of a convex octagon.

(n - 2) 180°

= (8 - 2) 180°

= 1080°

An octagon has 8 sides. So, substitute 8 for n.

Polygon ∕ Sum thm.

Simplify.

CONFIDENTIAL 15

Finding Interior Angle Measures and Sums in Polygons

B) Find the measure of each interior angle of a regular nonagon.

(n - 2) 180°

= (9 - 2) 180°

= 1260°

Substitute 9 for n.

Polygon ∕ Sum thm.

Simplify.

Step1: Find the sum of the interior angle measures.

Step2: Find the measure of one interior angle.

1260° = 140° 9

The int. ∕s are congruent, so divide by 9.

CONFIDENTIAL 16

Finding Interior Angle Measures and Sums in Polygons

C) Find the measure of each interior angle of a quadrilateral PQRS.

(4 - 2) 180° = 360°

m∕P + m∕Q + m∕R + m∕S = 360°

c + 3c + c + 3c = 360°

8c = 360° => c = 45°

Polygon ∕ Sum thm.

Polygon ∕ Sum thm.

Substitute.

P

Q R

Sc°

3c° c°

3c°

m∕P =m∕R = 45°

m∕Q = m∕S = 360°

CONFIDENTIAL 17

Now you try!

3a) Find the sum of the interior angle measures of a convex 15 - gon.

3b) Find the measure of each interior angle of a regular decagon.

CONFIDENTIAL 18

In the polygons below, an exterior angle has been measured at each vertex. Notice that in each case, the

sum of the exterior angle measure is 360°.

81°

132°147°

147° + 81° + 132° = 360°

41°55°

110°43°

111°

43° + 111° + 41° + 55° + 110° = 360°

CONFIDENTIAL 19

The sum of the exterior angle measures , one angle at each vertex, of a convex polygon with sides n is

360°.

Polygon Exterior Angle Sum Theorem

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Finding Exterior Angle Measures in Polygons

A) Find the measure of each exterior angle of a regular hexagon.

A regular hexagon has 6 ext ∕s. So, divide the sum by 6.

Polygon ext ∕ Sum thm.

A hexagon has 6 sides and 6 vertices.

Sum of the exterior angle = 360°

Measure of one exterior angle = 360° = = 60° 6

The measure of each exterior angle of a regular hexagon = 60°

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Finding Exterior Angle Measures in Polygons

B) Find the value of a in polygon RSTUV.

Combine like terms.

Polygon ext ∕ Sum thm. 7a° + 2a° + 3a° + 6a° + 2a° = 360°

20a°= 360°

a = 60°

2a°3a°

6a°2a°

7a°

R

ST

U

V

So, divide the sum by 20.

CONFIDENTIAL 22

Now you try!

4a) Find the sum of the measures of exterior angle of a regular dodecagon.

4b) Find the value of r in polygon JKLM.

J

K

L

M

4r°7r°

5r°8r°

CONFIDENTIAL 23

Photography ApplicationThe appearance of the camera is formed by ten blades. The blades overlap to form a regular decagon. What is

the measure of ∕CBD?

m ∕CBD = 360° =36° 10

A regular decagon has 10 congruent ext. angles. So, divide the sum by 10.

∕CBD is an exterior angle of a regular decagon. By the polygon exterior angle sum theorem, the sum of the exterior

measures is 360°. B

D

A

C

CONFIDENTIAL 24

Now you try!

5) Suppose the shutter of the camera were formed by 8 blades. What would the measure of each exterior

angle be?

CONFIDENTIAL 25

Now some problems for you to practice !

CONFIDENTIAL 26

Tell whether each figure is a polygon. If it is a polygon, name it by the number of its sides:

Assessment

1) 2)

CONFIDENTIAL 27

3) 4)

Tell whether each polygon is regular or irregular. Tell whether it is concave or convex:

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5) Find the measure of each interior angle of pentagon ABCDE.

A

B

C

D

E3x°

3x°

4x°

5x°

5x°

6) Find the measure of each interior angle of a regular dodecagon.

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7) Find the value of y in polygon JKLM.

8) Find the measure of each exterior angle of a regular pentagon.

J

K

L

M

4y°

4y° 2y°

6y°

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9) Name the polygon by the number of its sides.

10) In the polygon, /P, /R and /T are right angles and /Q is congruent to /S. What are m/Q and m/S?

R

P

Q S

T

CONFIDENTIAL 31

Today you will learn about the parts of polygon and the ways to classify polygons.

Each segment that forms a polygon is a side of the polygon. The common endpoint of two sides is a vertex of the

polygon. A segment that connects any two nonconsecutive vertices is a diagonal.

Properties and Attributes of Polygons

diagonal

A

E

B

CD

sidevertex

Let’s review

CONFIDENTIAL 32

You can name a polygon by the number of its sides. The table shows the names of some common polygons. Polygon ABCDE in the previous slide is a pentagon.

Number of Slides Name of Polygon

3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

9 Nonagon

10 Decagon

12 Dodecagon

n n - gon

CONFIDENTIAL 33

Identifying Polygon

Tell whether each figure is a polygon. If it is a polygon, name it by the number of its sides:

PolygonPentagon

Not a PolygonPolygonOctagon

CONFIDENTIAL 34

All the sides are congruent in an equilateral polygon. All the angles are congruent in an equiangular polygon. A regular polygon is one that is both equilateral and equiangular. If a

polygon is not regular, it is called irregular.

A polygon is concave if any part of a diagonal contains points in the exterior of the polygon. If no diagonal contains points in

the exterior, then the polygon is convex.

concave quadrilateral

convex quadrilateral

CONFIDENTIAL 35

The sum of the interior angle measures of a convex polygon with sides n is

(n - 2) 180°.

Polygon Angle Sum Theorem

CONFIDENTIAL 36

Finding Interior Angle Measures and Sums in Polygons

B) Find the measure of each interior angle of a regular nonagon.

(n - 2) 180°

= (9 - 2) 180°

= 1260°

Substitute 9 for n.

Polygon ∕ Sum thm.

Simplify.

Step1: Find the sum of the interior angle measures.

Step2: Find the measure of one interior angle.

1260° = 140° 9

The int. ∕s are congruent, so divide by 9.

CONFIDENTIAL 37

In the polygons below, an exterior angle has been measured at each vertex. Notice that in each case, the

sum of the exterior angle measure is 360°.

81°

132°147°

147° + 81° + 132° = 360°

41°55°

110°43°

111°

43° + 111° + 41° + 55° + 110° = 360°

CONFIDENTIAL 38

The sum of the exterior angle measures , one angle at each vertex, of a convex polygon with sides n is

360°.

Polygon Exterior Angle Sum Theorem

CONFIDENTIAL 39

Photography ApplicationThe appearance of the camera is formed by ten blades. The blades overlap to form a regular decagon. What is

the measure of ∕CBD?

m ∕CBD = 360° =36° 10

A regular decagon has 10 congruent ext. angles. So, divide the sum by 10.

∕CBD is an exterior angle of a regular decagon. By the polygon exterior angle sum theorem, the sum of the exterior

measures is 360°. B

D

A

C

CONFIDENTIAL 40

You did a You did a great great job today!job today!