1. 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.

2. Standards SPI’s taught in Unit 6:
SPI Give precise mathematical descriptions or definitions of geometric shapes in the plane and space.
SPI Determine areas of planar figures by decomposing them into simpler figures without a grid.
SPI Use coordinate geometry to prove characteristics of polygonal figures.
SPI Identify, describe and/or apply the relationships and theorems involving different types of triangles, quadrilaterals and other polygons.
SPI 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 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 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 Describe the properties of regular polygons, including comparative classification of them and special points and segments.
CLE Generate formulas for perimeter, area, and volume, including their use, dimensional analysis, and applications.
CFU (Checks for Understanding) applied to Unit 6:
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).
Recognize the capabilities and the limitations of calculators and computers in solving problems.
Describe and recognize minimal conditions necessary to define geometric objects.
Classify triangles, quadrilaterals, and polygons (regular, non-regular, convex and concave) using their properties.
Identify and apply properties and relationships of special figures (e.g., isosceles and equilateral triangles, family of quadrilaterals, polygons, and solids).
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.
Derive and use the formulas for the area and perimeter of a regular polygon. (A=1/2ap)

3. 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 B B B A C A A C C E D E D E D

4. 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
Name 3
Triangle 4
Quadrilateral 5
Pentagon 6
Hexagon 8
Octagon 9
Nonagon 10
Decagon 12
Dodecagon n
n-gon

5. Names of Polygons Generally accepted names
Other names not normally used
Sides
Name n
N-gon 3
Triangle 4
Quadrilateral 5
Pentagon 6
Hexagon 7
Heptagon 8
Octagon 10
Decagon 12
Dodecagon
Sides
Name 9
Nonagon, Enneagon 11
Undecagon, Hendecagon 13
Tridecagon, Triskaidecagon 14
Tetradecagon, Tetrakaidecagon 15
Pentadecagon, Pentakaidecagon 16
Hexadecagon, Hexakaidecagon 17
Heptadecagon, Heptakaidecagon 18
Octadecagon, Octakaidecagon 19
Enneadecagon, Enneakaidecagon 20
Icosagon 30
Triacontagon 40
Tetracontagon 50
Pentacontagon 60
Hexacontagon 70
Heptacontagon 80
Octacontagon 90
Enneacontagon
100
Hectogon, Hecatontagon
1,000
Chiliagon
10,000
Myriagon

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

7. 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

8. 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

9. 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

10. 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 = 360 1 2 5 4 3

11. 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?

12. 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 = degrees
The generic form of this equation is this:
Sum = [(n-2)*180]/n

13. Regular Polygons Name n -Sides (n-2) (n-2)*180 [(n-2)*180]/n Triangle
Total Interior
Angles
Each
Interior Angle
Name
n -Sides
(n-2)
(n-2)*180
[(n-2)*180]/n
Triangle 3 1
180 60
Quadrilateral 4 2
360 90
Pentagon 5
540
108
Hexagon 6
720
120
Octagon 8
1080
135
Nonagon 9 7
1260
140
Decagon 10
1440
144
Dodecagon 12
1800
150
n-gon n
n-2
Heptagon

14. Assignment Page 356 7-25 (guided practice)
Worksheet 3-4

15. Unit 6 Quiz 1
What is the sum of the measure of the interior angles of a 21-gon?
What is the sum of the measure of the interior angles of a 18-gon?
What is the sum of the measure of the interior angles of a 99-gon?
What is the sum of the measure of the interior angles of a 55-gon?
What is the measure of one interior angle of a 17-gon?
What is the measure of one interior angle of a 28-gon?
What is the equation used to solve the sum of the measure of the interior angles of a polygon? (all angles added together)
What is the name of a polygon with 12 sides?
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
Calculate the measure of one exterior angle to a regular Pentagon

16. 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

17. Review What x, and what is the measure of the missing angle? x 129
140 23 y 75
X-25 z
X-30 z x 75
140 y
X+15 45
X+25
150

18. Parallelogram a quadrilateral –has 4 sides Has 4 verticesb c d
a quadrilateral –has 4 sides
Has 4 vertices
Sum of interior angles is 360 degrees
Sum of Exterior angles is 360 degrees
Has both pairs of opposite sides parallel
Both pairs of opposite angles are congruent
Both pairs of opposite sides are congruent
Diagonals bisect each other
If one pair of opposite sides are congruent and parallel, then it is a parallelogram
In a parallelogram, consecutive angles are supplementary –as we reviewed

19. 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

20. Example using Consecutive Angles
Find the measure of angle C
Find the measure of angle B
Find the measure of angle A
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
B C
A D
Angle D + Angle A = 180, Angle D = 112, therefore Angle A = 180 – 112, or 680
1120
Opposite corners of a parallelogram have equal measure
680
Note that Angle A and C are equal, and Angle B and D are equal, and we’ve just proved why
680
1120

21. Example with Algebra Find the value of X in ABCD
Then find the length of BC and AD
Since opposite sides are congruent, set the values equal to each other
3X – 15 = 2X + 3
3X = 2X + 18
X = 18
B C
A D
3x - 15
If X = 18, then 3X – 15 = 39
If BC = 39, then AD = 39, since opposite sides are congruent
2x + 3

22. Another Algebra Example
Find the value of Y
Then find the measure of all angles
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
B C
A D
(3y + 37)
6y + 4

23. 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
3Y – X
X Y E

24. Another Algebra Example
Solve for m and n
m = n + 2
n + 10 = 2(n+2) – 8
n + 10 = 2n + 4 – 8
n + 10 = 2n – 4
n = 14
m =
m = 16
B C
A D
n m - 8
m n + 2

25. 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 H J K
We could draw a new transversal…
And we know the segments it makes are congruent to each other as well  HJ = JK

26. Assignment Page 364 9-27 (guided practice)
Worksheet 6-2 (independent practice)
Worksheet 6-3 (independent practice)

27. 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: points = 15
Quadrilateral –four sides
Has four vertices -corners
Opposite corners congruent
Opposite sides parallel
Consecutive corners supplementary
Diagonals bisect each other
Opposite sides congruent
Interior angle sum is 360
Exterior angle sum is 360
If it’s a regular quadrilateral, then it has equal angles and sides. Or –it’s normally not a regular quadrilateral
It’s a polygon

28. Answers to Quiz a quadrilateral –has 4 sides Has 4 vertices
Sum of interior angles is 360 degrees
Sum of Exterior angles is 360 degrees
Has both pairs of opposite sides parallel
Both pairs of opposite angles are congruent
Both pairs of opposite sides are congruent
Diagonals bisect each other
If one pair of opposite sides are congruent and parallel, then it is a parallelogram
In a parallelogram, consecutive angles are supplementary –as we reviewed

29. 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)

30. 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

31. 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 they
are on the perpendicular bisector made by
the diagonal used to connect them R
P Q S

32. Finding Angle Measure Example
MNPQ is a rhombus
Find the measure of the numbered angles
Angle 1 = Angle 3
Angles =180
2 x Angle 1 =
2 x Angle 1 = 60
Angle 1 = 30
Angle 3 = 30
Angle 2 and 4 = 30
N P
M Q 3 4
1200 1 2

33. 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 3

34. 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

35. 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 too
Why?

36. 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).

37. Assignment
Page odd (guided practice)
Worksheet 6-4

38. Unit 6 Quiz 3 Name 2 corresponding angles
Are corresponding angles congruent or supplementary?
Name 2 same side interior angles
Are same side interior angles congruent or supplementary?
Name 2 alternate interior angles
Are alternate interior angles congruent or supplementary?
If angle C is 70 degrees, what is the measure of angle E?
If angle B is 120 degrees, what is the measure of angle F?
If angle D is 125 degrees, what is the measure of angle E?
If angle E is 130 degrees, what is the measure of angle F? A B C D E F G H

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

40. 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

41. 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)
Angle 2 = 1800, therefore Angle 2 = 580 B
320 3
A C D

42. 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

43. Name the Quadrilateral
What is this?
Trapezoid
Parallelogram
Square
Rectangle
Rhombus

44. Classifying Quadrilaterals
These quadrilaterals that have both pairs of opposite sides parallel
Parallelograms
Rectangles
Rhombuses
Squares
These quadrilaterals that have four right angles
These quadrilaterals that have one pair of parallel sides
Trapezoid
Isosceles Trapezoid
These quadrilaterals have two pairs of congruent adjacent sides
Kites

45. Assignment Page 394-95 7-24, 28-36 Worksheet 6-5 6-1
Trapezoid Worksheet

46. 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

47. Example Polygon ABCD is congruent to polygon EFGH
Notice that the vertices that match each other are named in the same order
Imagine a mirror here A E B F C G D H

48. 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 = 100 degrees A L
135 E P B M
125 C N D O

49. 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
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?
C F
B E

50. 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 D B C

51. Assignment
Worksheet 4-1

52. Unit 6 Quiz 5

53. 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

54. 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

55. Example If X/Y = 5/6, Then? 6X = ? 5Y Y/X = ? 6/5 X/5 = ? Y/6
11/6
There is always a pattern…

56. Assignment
Page
Page

57. 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."

58. 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.

59. 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 A
127 E H EH B C
1270 53 A D BC

60. Determining Similarity
Determine whether the triangles are similar. If they are, write a similarity statement and give the similarity ratio
All three pairs of angles are congruent.
Check for proportionality of corresponding angles:
AC/FD = 18/24 = 3/4
AB/FE = 15/20 = 3/4
BC/ED = 12/16 = 3/4
Triangle ABC~Triangle FED with a similarity ratio of 3/4 or 3 : 4. B 15 12 A 18 C E 16 20 F D 24

61. Using Similarity LMNO~QRST Find the value of X LM/QR = ON/TS 5/6 = 2/X
Using this figure, find SR to the nearest tenth.
3.2 L 5 M T X S
5/6 = 3.2/SR
5(SR) = 6x3.2
SR = 6x3.2/5
SR = 3.8 6 Q R

62. Golden Rectangle / Ratio
Numbers…
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.
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 ( ). 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/30
X = 30/1.618
X = cm wide

63. ASSIGNMENT Page 475 2-7 (skip 4) Page 476 15-21

64. Unit 6 Final Extra Credit
Solve for the missing variables (2 points each, show work as required)
A=_______
B=_______
C=_______
D=_______
E=_______
5*a
9*d
Isosceles
Trapezoid
650
13.5*e
3*b+9
= 115, 115/5 = 23
3b = 24, 24/3 = 8
Side a = 12, side b = 12, side c = =17
360/8 = 45 , 45/9 = 5
135 = 13.5e , e = 10
Regular Octagon
c (round to nearest
whole number) 12 33
Right
Triangle
Rectangle