Unit 6 Introduction to Polygons

• This unit introduces Polygons. • It defines polygons and regular polygons, and has the Polygon

Angle Sum theorem. • This unit also details quadrilaterals, special quadrilaterals,

congruent polygons, similar polygons, and the Golden Ratio.

Standards• SPI’s taught in Unit 6:• SPI 3108.1.1 Give precise mathematical descriptions or definitions of geometric shapes in the plane and space. • SPI 3108.1.2 Determine areas of planar figures by decomposing them into simpler figures without a grid. • SPI 3108.3.2 Use coordinate geometry to prove characteristics of polygonal figures. • SPI 3108.4.3 Identify, describe and/or apply the relationships and theorems involving different types of triangles, quadrilaterals and other polygons. • SPI 3108.4.7 Compute the area and/or perimeter of triangles, quadrilaterals and other polygons when one or more additional steps are required

(e.g. find missing dimensions given area or perimeter of the figure, using trigonometry). • SPI 3108.4.12 Solve problems involving congruence, similarity, proportional reasoning and/or scale factor of two similar figures or solids. • CLE (Course Level Expectations) found in Unit 6:• CLE 3108.1.5 Recognize and use mathematical ideas and processes that arise in different settings, with an emphasis on formulating a problem in

mathematical terms, interpreting the solutions, mathematical ideas, and communication of solution strategies. • CLE 3108.4.2 Describe the properties of regular polygons, including comparative classification of them and special points and segments. • CLE 3108.4.6 Generate formulas for perimeter, area, and volume, including their use, dimensional analysis, and applications. • CFU (Checks for Understanding) applied to Unit 6:• 3108.1.5 Use technology, hands-on activities, and manipulatives to develop the language and the concepts of geometry, including specialized

vocabulary (e.g. graphing calculators, interactive geometry software such as Geometer’s Sketchpad and Cabri, algebra tiles, pattern blocks, tessellation tiles, MIRAs, mirrors, spinners, geoboards, conic section models, volume demonstration kits, Polyhedrons, measurement tools, compasses, PentaBlocks, pentominoes, cubes, tangrams).

• 3108.1.7 Recognize the capabilities and the limitations of calculators and computers in solving problems. • 3108.4.4 Describe and recognize minimal conditions necessary to define geometric objects.• 3108.4.9 Classify triangles, quadrilaterals, and polygons (regular, non-regular, convex and concave) using their properties. • 3108.4.10 Identify and apply properties and relationships of special figures (e.g., isosceles and equilateral triangles, family of quadrilaterals,

polygons, and solids). • 3108.4.12 Apply the Angle Sum Theorem for polygons to find interior and exterior angle measures given the number of sides, to find the number of

sides given angle measures, and to solve contextual problems. • 3108.4.28 Derive and use the formulas for the area and perimeter of a regular polygon. (A=1/2ap)

Polygons

• A polygon is a closed plane figure with at least three sides that are segments

• The sides of a polygon must intersect only at the endpoints. They cannot cross.

• To name a polygon start at any vertex (corner) and go in order around the polygon, either clockwise or counter clockwise

A

B

C

DE

AB

C

DE

A

B

C

D E

Classifying Polygons

• Convex Polygon: No vertex is “in” -- all point out

• Concave Polygon: Has at least one vertex “inside” – and two sides go in to form it

Sides Name3 Triangle4 Quadrilateral5 Pentagon6 Hexagon8 Octagon9 Nonagon

10 Decagon12 Dodecagonn n-gon

Names of PolygonsGenerally accepted names Other names not normally used

Sides Namen N-gon3 Triangle4 Quadrilateral5 Pentagon6 Hexagon7 Heptagon8 Octagon

10 Decagon12 Dodecagon

Sides Name9 Nonagon, Enneagon

11 Undecagon, Hendecagon13 Tridecagon, Triskaidecagon14 Tetradecagon, Tetrakaidecagon15 Pentadecagon, Pentakaidecagon16 Hexadecagon, Hexakaidecagon17 Heptadecagon, Heptakaidecagon18 Octadecagon, Octakaidecagon19 Enneadecagon, Enneakaidecagon20 Icosagon30 Triacontagon40 Tetracontagon50 Pentacontagon60 Hexacontagon70 Heptacontagon80 Octacontagon90 Enneacontagon

100 Hectogon, Hecatontagon1,000 Chiliagon

10,000 Myriagon

Polygon Angle Sum Theorem

• The sum of the measure of the interior angles of an n-gon is (n-2)*180

Name n -Sides (n-2) (n-2)*180Triangle 3 1 180Quadrilateral 4 2 360Pentagon 5 3 540Hexagon 6 4 720Octagon 8 6 1080Nonagon 9 7 1260Decagon 10 8 1440Dodecagon 12 10 1800n-gon n n-2 (n-2)*180

Example

• Find the sum of the measure of the interior angles of a 15-gon

• (By the way, if you have a cool calculator, this is where you turn open “apps”, then “A+ Geom”, then “A. Polygons” then enter 15 for number of sides)

• Sum = (n-2)* 180, or (15-2)* 180 or (13) * 180• Therefore, the sum of the interior angles is

2340

Example

• What if you are not told the number of sides, but are only told that the sum of the measure of the angles is 720? Can you determine the number of sides?

• If you have the cool calculator then use it now• Otherwise, substitute into the equation:• (n-2)*180 = 720, so• (n-2) = 720/180• n-2 = 4• n = 6

Example

• Find the measure of angle y• This is a 5 sided object• The sum of the interior angles is (n-2)*180 =

540 degrees• Therefore, we have 540 – 90 – 90 – 90 – 136 =

Y• So Y = 134 136

Y

Polygon Exterior Angle-Sum Theorem

• The sum of the measure of the exterior angles of a polygon, one at each vertex, is 360

• ALWAYS• It DOESN’T MATTER HOW MANY SIDES THERE

ARE, IT IS ALWAYS 360 DEGREES• Angle 1 + 2 + 3 + 4 + 5 = 360 1

2

34

5

Regular Polygons

• An Equilateral polygon has all sides equal• An Equiangular polygon has all angles equal• A REGULAR Polygon has all sides and all

angles equal –it is both equilateral and equiangular

• What are some examples of Regular Polygons in the real world?

Example

• If you have a regular polygon, then you can determine the measure of each interior angle

• For example, determine the measure of the sum of the interior angles of a regular 11-gon, and the measure of 1 angle

• Sum = (11-2)*180 = 1620• Since all angles are exactly the same, we can divide our

answer by the number of angles to find one angle• 1620/11 = 147.27 degrees• The generic form of this equation is this: • Sum = [(n-2)*180]/n

Regular PolygonsName n -Sides (n-2) (n-2)*180 [(n-2)*180]/n

Triangle 3 1 180 60Quadrilateral 4 2 360 90Pentagon 5 3 540 108Hexagon 6 4 720 120Octagon 8 6 1080 135Nonagon 9 7 1260 140Decagon 10 8 1440 144Dodecagon 12 10 1800 150n-gon n n-2 (n-2)*180 [(n-2)*180]/n

Total InteriorAngles

EachInterior Angle

Heptagon 7 5 900 128.6

Assignment

• Page 356 7-25 (guided practice)• Page 357 29-36 (guided practice)• Worksheet 3-4

Unit 6 Quiz 11. What is the sum of the measure of the interior angles of a 21-gon?2. What is the sum of the measure of the interior angles of a 18-gon?3. What is the sum of the measure of the interior angles of a 99-gon?4. What is the sum of the measure of the interior angles of a 55-gon?5. What is the measure of one interior angle of a 17-gon?6. What is the measure of one interior angle of a 28-gon?7. What is the equation used to solve the sum of the measure of the

interior angles of a polygon? (all angles added together)8. What is the name of a polygon with 12 sides?9. Given: A REGULAR Pentagon has 5 sides, and the sum of the measure of

the interior angles is 540 degrees. What equation would you use to find the measure of ONE angle

10. Calculate the measure of one exterior angle to a regular Pentagon

Classifying Quadrilaterals• This is what we already know about Quadrilaterals:• Four sides• Four corners –vertices• Sum of interior angles is 360 degrees• Sum of exterior angles is 360 degrees• If it is a “regular” quadrilateral, then each interior

angle is 90 degrees, and each exterior angle is 90 degrees and each side is equal in length

• Now we will begin to look at some Special Quadrilaterals

Review• What x, and what is the measure of the

missing angle?x

23

129

y

X+10

75

X+20

X-25

140

z

x 75 140 y X+15 45 X+25

X-30

150 z

1 2 3 4 5

6 7 8 9 10

Parallelogram1. a quadrilateral –has 4 sides 2. Has 4 vertices3. Sum of interior angles is 360 degrees4. Sum of Exterior angles is 360 degrees5. Has both pairs of opposite sides parallel6. Both pairs of opposite angles are congruent7. Both pairs of opposite sides are congruent8. Diagonals bisect each other9. If one pair of opposite sides are congruent and

parallel, then it is a parallelogram10. In a parallelogram, consecutive angles are

supplementary –as we reviewed

a

b

c

d

Consecutive Angles• Angles of a polygon that share a side are consecutive angles• For example, angle A and angle B share segment AB. Therefore they are

consecutive angles. – Which makes sense, because consecutive means “in order” and they

are “in order” on the polygon shown• On a parallelogram, consecutive angles are Same Side Interior angles,

which means they are supplementary• These angles are supplementary:• A and B• B and C• C and D• D and A

B C

A D

Example using Consecutive Angles• Find the measure of angle C• Find the measure of angle B• Find the measure of angle A

B C

A D

1120

Angle D + Angle C = 180, Angle D = 112, therefore Angle C = 180 – 112, or 680

Angle B + Angle C = 1800, Angle C = 68, therefore Angle B = 180 – 68, or 1120

Angle D + Angle A = 180, Angle D = 112, therefore Angle A = 180 – 112, or 680

Note that Angle A and C are equal, and Angle B and D are equal, and we’ve just proved why

1120

680

680Opposite corners of a parallelogram have equal measure

Example with Algebra

• Find the value of X in ABCD• Then find the length of BC and AD

B C

A D

3x - 15

2x + 3

• Since opposite sides are congruent, set the values equal to each other

• 3X – 15 = 2X + 3• 3X = 2X + 18• X = 18

• If X = 18, then 3X – 15 = 39• If BC = 39, then AD = 39,

since opposite sides are congruent

Another Algebra Example

• Find the value of Y• Then find the measure of all angles

B C

A D

(3y + 37)

6y + 4

• Since opposite angles are equal in a parallelogram, then set the values equal

• 3y + 37 = 6y + 4• 37 = 3y + 4• 3y = 33• y = 11• If y = 11, then • angle A = 6(11) + 4, or• angles A and C = 70• Angle B and D = 110

An Example with Algebra• Find the value of X and Y, and the value of AE, CE, BE, and DE• Set each side (value) equal to each other • Y = X + 1• 3Y – 7 = 2X• Choose a value to substitute for (we’ll use Y)• Therefore 3 (X + 1) – 7 = 2X• 3X + 3 – 7 = 2X• 3X – 4 = 2X• X = 4• Now solve for Y• 3Y – 7 = 2(4)• 3Y = 7 + 8• 3Y = 15• Y = 5• AE = 3(5) – 7, or 8• CE = AE, or 8• DE = Y, or 5• BE = DE, or 5

B C

A D

X + 1 Y3Y – 7

2

X

E

Another Algebra Example

• Solve for m and n

B C

A D

n + 10

2m - 8m n + 2

• m = n + 2• n + 10 = 2(n+2) – 8• n + 10 = 2n + 4 – 8• n + 10 = 2n – 4• n = 14• m = 14 + 2• m = 16

Transversal Theorem• If three (or more) parallel lines cut off

congruent segments on one transversal, then they cut off congruent segments on every transversal

• BD = DF, therefore• AC = CE A B

C D

E F

• We could draw a new transversal…

• And we know the segments it makes are congruent to each other as well HJ = JK

H

J

K

Assignment

• Page 364 9-27 (guided practice)• Page 365 29,30,38-40 (guided practice)• Page 372 7-15• Worksheet 6-2 (independent practice)• Worksheet 6-3 (independent practice)

Unit 6 Quiz 2

• There are ten (or more) characteristics of a parallelogram

• Name five of the ten characteristics (2 points each)

• 2 points each question (10 points)• 1 point extra credit for each additional

characteristic• Total possible score: 10 + 5 points = 15

Answers to Quiz1. a quadrilateral –has 4 sides 2. Has 4 vertices3. Sum of interior angles is 360 degrees4. Sum of Exterior angles is 360 degrees5. Has both pairs of opposite sides parallel6. Both pairs of opposite angles are congruent7. Both pairs of opposite sides are congruent8. Diagonals bisect each other9. If one pair of opposite sides are congruent and

parallel, then it is a parallelogram10. In a parallelogram, consecutive angles are

supplementary –as we reviewed

Narcissist

• A person who is overly self-involved, and often vain and selfish.

• Deriving gratification from admiration of his or her own physical or mental attributes.

• See also: “Nolen” (named changed to protect the innocent)

Rhombus

• Rhombus: a parallelogram with four congruent sides

• We can draw the same conclusions about Same Side Interior angles here –they are also corresponding angles, and any 2 in a row add up to 180 degrees (supplementary angle pairs)

Or

Rhombus Theorems

• Each diagonal of a rhombus bisects 2 angles of the rhombus• The diagonals of a rhombus are perpendicular• If we remember the Perpendicular Bisector theorem, we

know that if 2 points are equally distant from the endpoints of a line segment, then they are on the perpendicular bisector. That is the case here.Points R and S are equally distant from points P and Q. Therefore theyare on the perpendicular bisector made bythe diagonal used to connect them

R

P Q

S

Finding Angle Measure Example

• MNPQ is a rhombus• Find the measure of the numbered angles• Angle 1 = Angle 3• Angles 1 + 3 + 120 =180• 2 x Angle 1 = 180-120• 2 x Angle 1 = 60• Angle 1 = 30• Angle 3 = 30• Angle 2 and 4 = 30

N P

M Q

12

34

1200

Another Find the Measure Example

• What is the measure of Angle 2?• 50 degrees (alternate interior angles are equal)• What is the measure of Angle 3?• 50 degrees (the diagonal is an angle bisector, so if angle 2 is

50 degrees, angle 3 is 50 degrees)• What is the measure of Angle 1?• 90 degrees (the diagonals of a rhombus are perpendicular

bisectors)• What is the measure of Angle 4?• 40 degrees (180 degrees minus 90 degrees

minus 50 degrees)

500

1

4 23

Rectangle

• Rectangle: a parallelogram with four right angles

• Diagonals on a rectangle are equal• Note: All four sides do not have to be equal,

but opposite sides are (because it’s a parallelogram)

Or

Square

• Square: a parallelogram with four congruent sides and four right angles

• Is a square a rectangle? Is a rectangle a square?

NOTE: Diagonals on a square are equal tooWhy?

Check on Learning• The quadrilateral has congruent diagonals and one angle of 600. Can it be

a parallelogram?• No. A parallelogram with congruent diagonals is a rectangle with four 900

angles.• The quadrilateral has perpendicular diagonals and four right angles. Can it

be a parallelogram?• Yes. Perpendicular diagonals means that it is a rhombus, and four right

angles means it would be a rectangle. Both properties together describe a square.

• A diagonal of a parallelogram bisects two angles of the parallelogram. Is it possible for the parallelogram to have sides of lengths 5,6,5 and 6?

• No. If a diagonal (of a parallelogram) bisects two angles then the figure is a rhombus, and rhombuses have all sides the same size (congruent).

Assignment

• Page 379-80 7-39 odd (guided practice)• Worksheet 6-4

Unit 6 Quiz 31. Name 2 corresponding angles2. Are corresponding angles congruent or

supplementary?3. Name 2 same side interior angles4. Are same side interior angles congruent or

supplementary?5. Name 2 alternate interior angles6. Are alternate interior angles congruent or

supplementary?7. If angle C is 70 degrees, what is the measure of

angle E?8. If angle B is 120 degrees, what is the measure of

angle F?9. If angle D is 125 degrees, what is the measure of

angle E?10. If angle E is 130 degrees, what is the measure of

angle F?

A B

E F

C D

HG

Kite

• Kite: a quadrilateral with two pairs of adjacent sides congruent, and no opposite sides congruent

Kites

• Remember, a Kite is a quadrilateral with two pairs of adjacent sides congruent and no opposite sides congruent.

• The diagonals of a kite are perpendicular

Perpendicular

Example –Find a Measure of an Angle in a Kite

• Find the measure of Angle 1, 2 and 3

• Angle 1 = 90 degrees (diagonals of a kite are perpendicular)

• 900 + 320 + Angle 2 = 1800, therefore Angle 2 = 580

1 2

3 320

A C

B

D

Trapezoid

• Trapezoid: a quadrilateral with exactly one pair of parallel sides.

• The isosceles trapezoid is one whose nonparallel opposite sides are congruent

• Again, we can conclude Supplementary Angles

On an Isosceles trapezoid, the diagonals are congruent

Name the Quadrilateral

1. What is this?

2. What is this?

3. What is this?

4. What is this?

5. What is this?

Trapezoid

Parallelogram

Square

Rectangle

Rhombus

Classifying Quadrilaterals• These quadrilaterals that have both pairs of opposite sides

parallel– Parallelograms

• Rectangles• Rhombuses• Squares

• These quadrilaterals that have four right angles– Squares– Rectangles

• These quadrilaterals that have one pair of parallel sides– Trapezoid– Isosceles Trapezoid

• These quadrilaterals have two pairs of congruent adjacent sides– Kites

Assignment

• Page 394-95 7-24, 28-36• Worksheet 6-5 6-1• Trapezoid Worksheet

Congruent Polygons• Congruent Figures have the same size and shape• When figures are congruent, it is possible to move

one over the second one so that it covers it exactly• Congruent polygons have congruent corresponding

parts –the sides and angles that match up are exactly the same

• Matching vertices (corners) are corresponding vertices. When naming congruent polygons, always list the corresponding vertices in the same order

Example

• Polygon ABCD is congruent to polygon EFGH• Notice that the vertices that match each other

are named in the same order

A

D

B

C

E

F

G

H

Imagine a mirror here

Example • Polygon ABCDE is congruent to Polygon LMNOP• Determine the value of angle P

• Using the Polygon Angle Sum Theorem, we know that a 5 sided polygon has (5-2) 180, or 540 total interior degrees∙

• Because the polygons are congruent, we know that Angle B is congruent to Angle M. To solve for Angle P, we take 540 -90 -90 -125 -135 = 100 degrees

A L

B M

C ND O

E P125

135

Congruent Triangles

• We are going to learn many, many ways to prove triangles are congruent

• Here is the first part of many of these proofs:• If two angles of one triangle are congruent to

two angles of another triangle, then the third angles are congruent

A D

C FB E

If Angle A is congruent to Angle D, and Angle C is congruent to Angle F, then Angle B is congruent to Angle E. Why?

Example• Is triangle ABC congruent to triangle

ADC?• We need all 3 sides congruent, and all 3

angles congruent• Side AD is congruent to side AB• Side DC is congruent to side CB• Side AC is congruent to itself• Angle DAC is congruent to Angle BAC• Angle ADC is congruent to Angle ABC• All we need is the 3rd angle –Angle DCA

must be congruent to Angle BCA• Since we have 2 angles congruent, we

know the 3rd angle is congruent

A

B

C

D

Assignment

• Worksheet 4-1

Unit 6 Quiz 5

Ratios and Proportions

• Proportion: a statement that two ratios are equal. It can be written either as

• A/B = C/D• A:B = C:D• Extended proportion: When 3 or more ratios

are equal, such as 6/24 = 4/16 = 1/4

Properties of Proportions

• A/B = C/D is equivalent to• 1) AD = BC (cross-product Property)

– Also written A:B = C:D, it can be referred to the “Product of the extremes (the two outside numbers) is equal to the product of the means (the two inside numbers)”

• 2) B/A = D/C (flip both sides)• 3) A/C = B/D (ratio of the “tops” is equal to the ratio

of the “bottoms”• 4) (A + B)/ B = (C + D)/D

Example

• If X/Y = 5/6, Then?• 6X = ?• 5Y• Y/X = ?• 6/5• X/5 = ?• Y/6• (X + Y)/Y = ?• 11/6• There is always a pattern…

Assignment

• Page 461 6-9• Page 462 33-47

John Wayne as The Shootist

• "I won't be wronged, I won't be insulted, and I won't be laid a hand on. I don't do these things to other people and I expect the same from them."

Similar Polygons• Two figures that have the same shape but not

necessarily the same size are similar This is the symbol for similarity: ~

• Two polygons are similar if– 1) Corresponding angles are congruent and– 2) Corresponding sides are proportional.

• The ratio of the lengths of corresponding sides is the similarity ratio.

Example of Similarity

• If ABCD ~ EFGH then:• ‘m of angle E = m of angle ___• So m of E = m of A = 53 because

corresponding angles are congruent.• AB/EF = AD/___• So AB/EF = AD/EH because

corresponding sides are proportional.• We can conclude that the m of angle

B = _____• We can conclude that GH/CD =

FG/___

F G

E

B

H

A D

C

53

127 A

EH

1270

BC

Determining Similarity• Determine whether the triangles

are similar. If they are, write a similarity statement and give the similarity ratio

1. All three pairs of angles are congruent.

2. Check for proportionality of corresponding angles:

1. AC/FD = 18/24 = 3/42. AB/FE = 15/20 = 3/43. BC/ED = 12/16 = 3/4

• Triangle ABC~Triangle FED with a similarity ratio of 3/4 or 3 : 4.

A

F

E

D

C

B

18

1215

16 20

24

Using Similarity

• LMNO~QRST• Find the value of X• LM/QR = ON/TS• 5/6 = 2/X• 5X = 12• X = 2.4• Using this figure, find SR to the

nearest tenth.

L

T

Q

S

O N

M

R

2

3.2

5

X

6

5/6 = 3.2/SR5(SR) = 6x3.2SR = 6x3.2/5SR = 3.8

Golden Rectangle / Ratio

• A Golden Rectangle is a rectangle that can be divided into a square and a rectangle that is similar to the original rectangle.

• In any golden rectangle, the length and width are in the Golden Ratio, which is about 1.618 : 1.

• The golden rectangle is considered pleasing to the human eye. It has appeared in architecture and art since ancient times. It has intrigued artists including Leonardo da Vinci (1452 - 1519). Da Vinci illustrated The Divine Proportion, a book about the golden rectangle.

• An artist plans to paint a picture. He wants the canvas to be a golden rectangle, with the longer horizontal sides to be 30 cm wide. How high should the canvas be?

And so on…

1/1.618 = x/30X = 30/1.618X = 18.54 cm wide

Numbers…

ASSIGNMENT

• Page 475 2-7 (skip 4)• Page 476 15-21• Page 477 24-27, 29,30, 39-42

Unit 6 Final Extra Credit

• Solve for the missing variables (2 points each, show work as required)

IsoscelesTrapezoid

Rectangle

Regular Octagon

RightTriangle

5*a

650

3*b+9

3312c (round to nearest whole number)

13.5*e

9*d

1. A=_______2. B=_______3. C=_______4. D=_______5. E=_______