Transcript Unit 6 Introduction to Polygons

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.
B
B
B
A
C
E
D
A
E
A
C
C
D
D
E
• To name a polygon start at any vertex (corner)
and go in order around the polygon, either
clockwise or counter clockwise
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
3
4
5
6
8
9
10
12
n
Name
Triangle
Quadrilateral
Pentagon
Hexagon
Octagon
Nonagon
Decagon
Dodecagon
n-gon
Names of Polygons
Generally accepted names
Sides
n
3
4
5
6
7
8
10
12
Name
N-gon
Triangle
Quadrilateral
Pentagon
Hexagon
Heptagon
Octagon
Decagon
Dodecagon
Other names not normally used
Sides
9
11
13
14
15
16
17
18
19
20
30
40
50
60
70
80
90
100
1,000
10,000
Name
Nonagon, Enneagon
Undecagon, Hendecagon
Tridecagon, Triskaidecagon
Tetradecagon, Tetrakaidecagon
Pentadecagon, Pentakaidecagon
Hexadecagon, Hexakaidecagon
Heptadecagon, Heptakaidecagon
Octadecagon, Octakaidecagon
Enneadecagon, Enneakaidecagon
Icosagon
Triacontagon
Tetracontagon
Pentacontagon
Hexacontagon
Heptacontagon
Octacontagon
Enneacontagon
Hectogon, Hecatontagon
Chiliagon
Myriagon
Polygon Angle Sum Theorem
• The sum of the measure of the interior angles
of an n-gon is (n-2)*180
Name
Triangle
Quadrilateral
Pentagon
Hexagon
Octagon
Nonagon
Decagon
Dodecagon
n-gon
n -Sides
3
4
5
6
8
9
10
12
n
(n-2)
1
2
3
4
6
7
8
10
n-2
(n-2)*180
180
360
540
720
1080
1260
1440
1800
(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
136
• So Y = 134
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
5
4
3
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 Polygons
Name
n -Sides
Triangle
3
Quadrilateral
4
Pentagon
5
Hexagon
6
Octagon
8
Nonagon
9
Decagon
10
Dodecagon
12
n-gon
n
Heptagon
7
(n-2)
1
2
3
4
6
7
8
10
n-2
5
Total
Interior
Angles
Each
Interior
Angle
(n-2)*180 [(n-2)*180]/n
180
60
360
90
540
108
720
120
1080
135
1260
140
1440
144
1800
150
(n-2)*180 [(n-2)*180]/n
900
128.6
Assignment
• Page 356 7-25 (guided practice)
• Page 357 29-36 (guided practice)
• Worksheet 3-4
Unit 6 Quiz 1
1.
2.
3.
4.
5.
6.
7.
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)
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
129
23
1
X+10
y
75
2
3
X+20
X-25
4
140
z
5
X-30
x
6
75
140
7
y
X+15 45
8
X+25
9
150
10
z
1.
2.
3.
4.
5.
6.
7.
8.
9.
Parallelogram
a
a quadrilateral –has 4 sides
Has 4 vertices
b
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
10. In a parallelogram, consecutive angles are
supplementary –as we reviewed
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
B
supplementary:
• A and B
• B and C
• C and D
• D and A
A
D
C
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
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
A
680
B
1120
C
680
Opposite
corners of a
parallelogram
have equal
measure
1120
D
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 - 15
B
• 3X = 2X + 18
• X = 18
• If X = 18, then 3X – 15 = 39
• If BC = 39, then AD = 39,
since opposite sides are
congruent
A
2x + 3
D
C
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
B
C
(3y + 37)
• 3y = 33
• y = 11
• If y = 11, then
• angle A = 6(11) + 4, or
• angles A and C = 70
• Angle B and D = 110 6y + 4
A
D
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
B
3Y – 7 = 2(4)
3Y = 7 + 8
3Y = 15
Y=5
E
AE = 3(5) – 7, or 8
CE = AE, or 8
DE = Y, or 5
BE = DE, or 5
A
C
D
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 = 14 + 2
m = 16
A
B
C
D
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
A
B
H
• AC = CE
• We could draw a new
transversal…
• And we know the
segments it makes are
congruent to each other
as well  HJ = JK
C
E
J
K
D
F
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
1.
2.
3.
4.
5.
6.
7.
8.
9.
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
10. 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
Or
• 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)
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
R
from points P and Q. Therefore they
are on the perpendicular bisector made by
P
the diagonal used to connect them
S
Q
Finding Angle Measure Example
•
•
•
•
•
•
•
•
•
MNPQ is a rhombus
Find the measure of the numbered angles
Angle 1 = Angle 3
N
Angles 1 + 3 + 120 =180
1200
2 x Angle 1 = 180-120
2 x Angle 1 = 60
M 12
Angle 1 = 30
Angle 3 = 30
Angle 2 and 4 = 30
3
4
P
Q
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?
500
40 degrees (180 degrees minus 90 degrees
1
minus 50 degrees)
3
4
2
Rectangle
• Rectangle: a parallelogram with four right
angles
Or
• 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)
Square
• Square: a parallelogram with four congruent
sides and four right angles
NOTE: Diagonals
on a square are
equal too
Why?
• Is a square a rectangle? Is a rectangle a
square?
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 3
1.
2.
Name 2 corresponding angles
Are corresponding angles congruent or
supplementary?
3. Name 2 same side interior angles
4. Are same side interior angles congruent or
supplementary?
5. Name 2 alternate interior angles
6. 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
C D
E F
GH
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
B
320
3
A
1
D
2
C
Trapezoid
• Trapezoid: a quadrilateral with exactly one
pair of parallel sides.
On an Isosceles
trapezoid, the
diagonals are
congruent
• The isosceles trapezoid is one whose
nonparallel opposite sides are congruent
• Again, we can conclude Supplementary Angles
Name the Quadrilateral
1. What is this?
Trapezoid
2. What is this?
Parallelogram
3. What is this?
Square
4. What is this?
Rectangle
5. What is this?
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
Imagine a mirror here
A
E
B
C
D
F
G
H
Example
• Polygon ABCDE is congruent to Polygon LMNOP
• Determine the value of angle P
A
E
D
125 B
P
C
O
L
135
M
N
• 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
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
F
B
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
D
B
C
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
F
G
127
E
H
B
C
53
A
D
• If ABCD ~ EFGH then:
• ‘m of angle E = m of angle ___ A
• So m of E = m of A = 53 because
corresponding angles are congruent.
EH
• AB/EF = AD/___
• So AB/EF = AD/EH because
corresponding sides are proportional.
• We can conclude that the m of angle
0
B = 127
_____
• We can conclude that GH/CD =
BC
FG/___
Determining Similarity
•
B
15
A
12
18
C
E
16
D
20
24
F
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 = 3/4
2. AB/FE = 15/20 = 3/4
3. BC/ED = 12/16 = 3/4
•
Triangle ABC~Triangle FED with a
similarity ratio of 3/4 or 3 : 4.
Using Similarity
N
2
O
3.2
L
5
T
Q
M
X
6
S
R
•
•
•
•
•
•
•
LMNO~QRST
Find the value of X
LM/QR = ON/TS
5/6 = 3.2/SR
5/6 = 2/X
5(SR) = 6x3.2
5X = 12
SR = 6x3.2/5
X = 2.4
SR = 3.8
Using this figure, find SR to the
nearest tenth.
Golden Rectangle / Ratio
Numbers…
And so on…
1/1.618 = x/30
X = 30/1.618
X = 18.54 cm wide
• 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?
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)
1. A=_______
5*a
9*d
Isosceles
Trapezoid
650
Rectangle
B=_______
C=_______
D=_______
E=_______
13.5*e
Regular Octagon
12
2.
3.
4.
5.
c (round to nearest
Right
whole number)
Triangle