Download File - Ms. Brown`s class

Survey
yes no Was this document useful for you?
   Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Document related concepts

Steinitz's theorem wikipedia, lookup

Golden ratio wikipedia, lookup

Technical drawing wikipedia, lookup

Simplex wikipedia, lookup

Approximations of π wikipedia, lookup

Tessellation wikipedia, lookup

Regular polytope wikipedia, lookup

Perceived visual angle wikipedia, lookup

Multilateration wikipedia, lookup

Shapley–Folkman lemma wikipedia, lookup

Rational trigonometry wikipedia, lookup

Euler angles wikipedia, lookup

Complex polytope wikipedia, lookup

List of regular polytopes and compounds wikipedia, lookup

Trigonometric functions wikipedia, lookup

History of trigonometry wikipedia, lookup

Pythagorean theorem wikipedia, lookup

Euclidean geometry wikipedia, lookup

Integer triangle wikipedia, lookup

Compass-and-straightedge construction wikipedia, lookup

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
Twitter #polygonsproperties

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
Quadrilateral
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?



Think about it….
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:
AE = AD = ED
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.
 Take your time
 Good LUCK!
Practice





Find a partner.
Complete “How Can You Tell Which End of a Worm Is His
Head?” (must complete in pairs).
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.

3. Quadrilaterals ABCD and
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.