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Transcript
```Warm Up



Solve each proportion.
1. 3 = 15
x
20
3.
x
35
2.
3
4
24
y
4
7
5
ACTIVITY
30min
`
Materials

Need glue & scissors
Dissections of
Squares



Critical Thinking
Cut out each figure
(one at a time) and
rearrange pieces to
form a square.
Glue pieces in the
square provided.
Polygons and Similarities
6.1 and Chapter 7
6.1 Properties and Attributes of
Polygons
To identify and name polygons
 To find the sum of the measures of interior
and exterior angles of convex polygons and
measures of interior and exterior angles of
regular polygons
 To solve problems involving angle measures of
polygons

Define Polygons


A closed figure
Formed by segments
Polygons vs. Not Polygons
Regular Polygon

Convex polygon where all sides and angles are congruent.
Types of Polygons


A Convex polygon is a
polygon such that no lines
containing a side of the
polygon contains a point
in the interior of the
polygon
~Angles face out


A Concave polygon is a
polygon for which there is
a line containing a side of
the polygon and a point in
the interior of the
polygon.
~ Angles cave in
Convex or Concave
Possible Diagonals/Triangles:
Find the sum of degrees
Determine how many triangles a shape have by drawing
diagonals. Each triangle is 180 degrees.
Thm 6-1-1: Polygon Angle Sum Theorem



Formula: S = 180 (n – 2)
S= sum of the measures of interior angles
N = number of sides
Sum
S = 180(n -2)
S = 180(3-2)
S = 180(1)
S = 180
An interior angle I = S/n
I = 180/3
I = 60
Thm6-1-2: Polygon Exterior Angle Sum
Theorem





The sum of the measures of the exterior angles is 360°
To find one exterior angle use formula
Formula: E = 360/ n
E= measure of exterior angles
N = number of sides
E = 360/n
E = 360/3
E = 120
Warm-Up: Polygons

Number of sides: 6

Name Polygon: Hexagon

Number of Triangles:
(n- 2) = 4

Is polygon convex or concave.

Find sum of the measures of
interior angles:
S = 180(n-2)

180(4)= 720
Find the measure of an interior
angle
I = Sum
720 = 120
n

6
Find measure of an exterior angle:
E = 360
n
360 = 60
6

P. 397 Think and Discuss #1

Draw a concave pentagon
and convex pentagon.
Explain the difference
between the two figures.
Textbook p. 395
Convex Polygons
Number of Sides
Number of
triangles
Sum of Angles
Measures
S = 180(n – 2)
Measure of an
Exterior Angle
Ext = 360/n
Triangle
Pentagon
Hexagon
Heptagon
Octagon
Nonagon
Decagon
Hendecagon
Dodecagon
n - gon
n
(n – 2)
S = 180( n – 2)
Ext = 360 / n
Practice

Geo: Textbook page 398 # 1- 15
Warm-Up: Polygons

Number of sides: 7

Name Polygon: Heptagon

Number of Triangles:
(n- 2) = 5

Is polygon convex or concave.

Find sum of the measures of
interior angles:
S = 180(n-2)

180(5)= 900
Find the measure of an interior
angle
I = Sum
900 = 128.57
n

7
Find measure of an exterior angle:
E = 360
n
360 = 51.43
7
Warm Up: Part 2


Solve each proportion.
1. x = 11
2. 13 = 26
5 35
49 7x
3. x-2 = 3
x
8
Cross Multiply Cross Multiply
Cross Multiply
- 35(x) = 11(5) - 13(7x) = 49(26) - 8(x-2) = 3(x)
- 35x = 55
91x = 1274 - 8x -16 = 3x
- 35x = 55
35 35
- x = 1.57
- 91x = 1274
91
91
- x = 14
- 8x -8x -16 = 3x – 8x
-
-16 = -5x
-
-16 = -5x
-5 -5
- x = 3.2
Practice

Geo: Textbook page 398 # 1- 15
Chapter 7: Connecting Proportion and Similarity
 Recognize
and use ratios and
proportions
 Identify similar figures and use the
proportions of similar figures to solve
problems
 Use proportional parts of triangles to
solve problems
Exploring Similar Polygons



Book Definition:
Two polygons are similar
if and only if their
corresponding angles are
congruent and the
measures of their
corresponding sides are
proportional.
Symbol:


In simpler terms:
Two polygons with the
same shape but are
different sizes.
Are two congruent figures similar?



Discuss…
Yes, congruent figures have congruent angles and sides are
proportional at 1:1 ratio
Scale Factor and Dilation


Dilation is a
transformation that
reduces or enlarges
figures
On a camera: zoom-in/
zoom-out

Scale Factor = ______
A
5
D
2.5
4
B
3
C
10 =8 =6 = 5
5 4 3 2.5
E
10
H
5
dfgff
8
F
6
G
Scale Factor
Scale Factor = _____

12 =9 = 24
4 3
8
4
Scale Factor = ______
A
6
D
2
5
8
B
3
12
24
4
C
12 =8 =7 =4
6 5 4 2
E
12
H
4
dfgff
8
9
F
7
G
Similar Triangles: Property 1

Side-Side-Side Similarity
__________________________:
the three sides
if _____________________
of one triangle are
proportional
the three corresponding sides
____________
to __________________________
of
similar
another triangle, then the triangles are ____________.
 Ex.
Similar Triangles: Property 2

Angle - Angle Similarity
_________________________:
two angles of ___________
one triangle
If the measures of ___________
congruent to 2 angles of the other
are __________________________________
similar
then the triangles are __________.
Ex.
Similar Triangles: Property 3
Side-Angle-Side Similarity
________________________:
two sides of a triangle are ___________
proportional
if the measures of __________

two sides of another triangle and the
to the measures of _________
included angles are _________,
congruent then the triangles are
_________
similar
___________.
Ex.
Similarity Ratio:
Triangle Proportionality Theorem

VW = VX
WY XZ


Similarity Ratio:
AC AB
CB


Similarity Ratio:
EC = ED = CD
EA
EB
AB
X
Solve for x and y.


49
29
x


20
y+3
21
Warm- up:

1. Draw a convex heptagon.

2. Draw a concave nonagon

3. Sketch a regular hexagon.



Sum of interior angles: S = 180(6-2) = 720
I = 720/6 = 120
An Interior angle:
An Exterior angle:
E = 360/6 = 60
Check Practice: Similar Figures

Notes: Sides are proportional & angles are congruent.

P. 520 # 5, 6, 12, 13, 16, 17, 19
Quiz Time 10 mins
 Absolutely
NO TALKING during
quiz.
 Good LUCK!
Practice





Find a partner.
Complete “How Can You Tell Which End of a Worm Is His
Each partner complete a side
For each answer, write in letter of question. EACH SIDE
IS DIFFERENT!
Each partner must show work to receive FULL CREDIT.
Use notebook paper
Chapter 7: Practice
Warm-Up

1. Use the Interior Angle
Theorem to find measure of
each angle.
C
B
3x-12
4x-14

A 7x +5



2. Are two congruent figures
similar? Explain.

AEFG are similar. AE=22,
AB=5x + 4, EF=8 and
BC=2x-2. Solve for x.
A
E
B
5x -15 D



5x
E
G
D
F
C
Warm -Up
16
22
11 m
3
8
4
x
4
Warm up

1. Similar? Yes or No

a.
4
6
b.
7
6
8
2. What is the difference
between concave and
convex polygons?

3. regular heptagon, find
each.
A. Sum of I =
B. I=
C. E=
D. Sum of E=
1.8
1.4


1.3

15
20



5

12.5
Constructions


Materials: paper, ruler and protractor
Construct each (use polygon chart)







Regular Pentagon ~ sides 3 inches
Regular Octagon ~ sides 2 inches
Your Choice ~ Regular Polygon ~ make sides a reasonable
measure
To start… find the measure of one interior angle. Draw
one measured side with ruler.
Next use protractor to measure angle
Then draw 2nd side
Repeat steps.
Practice: Constructions

Construct each (use polygon chart)

Materials: paper, ruler and protractor
 Include the sum of interior angles and the measure of one
interior angle
 Regular Pentagon ~ sides 3 inches
 Regular Octagon ~ sides 2 inches
 Your Choice ~ Regular Polygon (4+ sides) ~ make sides a
reasonable measure
 Bonus: Your Choice ~ Regular Polygon (4+ sides) ~ make sides
a reasonable measure
To start… Draw one measured side with ruler.




Next find the measure of one interior angle and use
protractor to measure interior angle
Then draw 2nd side
Repeat steps.
Warm up

1. Similar? Yes or No

a.
4
6
b.
7
6
8
2. What is the difference
between concave and
convex polygons?

3. regular heptagon, find
each.
A. Sum of I =
B. I=
C. E=
D. Sum of E=
1.8
1.4


1.3

15
20



5

12.5
Bonus



1. Draw if needed. Joyce
sighted to the top of a tree
along a stake that she knew
to be 3 feet high. If she is
standing 2 feet from the stake
and 18 feet from the tree,
how high is the tree?
2. Find x.
2x+32
8x+5


10x+2
12x+3
4x+12
4x+2

3. The lengths of the sides of
a triangle are 3, 4, and 6. If
the length of the shortest side
of a similar triangle is 5, find
the lengths of its other two
sides.
```
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