similarity
DESCRIPTION
Similarity. When two objects are congruent , they have the same shape and size. Two objects are similar if they have the same shape, but different sizes. Their corresponding parts are all proportional . Any kind of polygon can have two that are similar to each other. Similarity. - PowerPoint PPT PresentationTRANSCRIPT
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When two objects are congruent, they have the same shape and size.
Two objects are similar if they have the same shape, but different sizes.
Their corresponding parts are all proportional.
Any kind of polygon can have two that are similar to each other.
When two objects are congruent, they have the same shape and size.
Two objects are similar if they have the same shape, but different sizes.
Their corresponding parts are all proportional.
Any kind of polygon can have two that are similar to each other.
SimilaritySimilarity
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Examples: 2 squares that have different lengths of sides.
2 regular hexagons
Examples: 2 squares that have different lengths of sides.
2 regular hexagons
SimilaritySimilarity
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Similar Polygons (7-2)Similar Polygons (7-2)
Characteristics of similar polygons:1. Corresponding angles are congruent
(same shape)
2. Corresponding sides are proportional
(lengths of sides have the same ratio)
ABCD ~ EFGH
Vertices must be listed in order when naming
Characteristics of similar polygons:1. Corresponding angles are congruent
(same shape)
2. Corresponding sides are proportional
(lengths of sides have the same ratio)
ABCD ~ EFGH
Vertices must be listed in order when naming
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Similar Polygons (7-2)Similar Polygons (7-2)
ABCD ~ EFGH
Complete the statements.
ABCD ~ EFGH
Complete the statements.
115
65
H
F
G
B
C D
A
E
AB
EF =
BC
?
HG
DC =
FG
?
mF = ?
mC = ?
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Similar Polygons (7-2)Similar Polygons (7-2)
Determine whether the parallelograms are similar. Explain.
Determine whether the parallelograms are similar. Explain.
44
2
2
22
1
1
114
59
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Similar Polygons (7-2)Similar Polygons (7-2)
Scale factor- ratio of the lengths of two corresponding sides of two similar polygons
The scale factor can be used to determine unknown lengths of sides
Scale factor- ratio of the lengths of two corresponding sides of two similar polygons
The scale factor can be used to determine unknown lengths of sides
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Similar Polygons (7-2)Similar Polygons (7-2)
If ABC ~ YXZ, find the scale factor of the large triangle to the small and find the value of x.
scale factor = 5/2 x= 16
If ABC ~ YXZ, find the scale factor of the large triangle to the small and find the value of x.
scale factor = 5/2 x= 16
x
12
A
BC
40
30Z
X
Y
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Example from Similar Polygons Worksheet
Example from Similar Polygons Worksheet
Are the two polygons shown similar? Corresponding angles must be congruent All pairs of corresponding sides must be
proportional (same scale factor)
Are the two polygons shown similar? Corresponding angles must be congruent All pairs of corresponding sides must be
proportional (same scale factor)
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Example from Using Similar Polygons Worksheet
Example from Using Similar Polygons Worksheet
Given two similar polygons. Find the missing side length. Redraw one of the polygons so corresponding
sides match up (if needed) Determine the scale factor Set up a proportion and solve for the missing
side length
Given two similar polygons. Find the missing side length. Redraw one of the polygons so corresponding
sides match up (if needed) Determine the scale factor Set up a proportion and solve for the missing
side length
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Similar Polygons (7-2)Similar Polygons (7-2)
Homework Similar Polygons worksheet #1-17
odd Using Similar Polygons worksheet
#1-15 odd
Homework Similar Polygons worksheet #1-17
odd Using Similar Polygons worksheet
#1-15 odd
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Scale DrawingScale Drawing
Problem 2 on p.443
Complete Similarity Application Problems
More practice
p.444 #9, 13, 15, 19, 23, and 25
Problem 2 on p.443
Complete Similarity Application Problems
More practice
p.444 #9, 13, 15, 19, 23, and 25
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Similar Triangles (7-3)Similar Triangles (7-3)
AA ~ Postulate – If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
AA ~ Postulate – If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
A' C'
B'
A
B
C
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Similar Triangles (7-3)Similar Triangles (7-3)
SAS ~ Theorem – If an angle of one triangle is congruent to an angle of a second triangle, and the sides including the two angles are proportional, then the triangles are similar.
SAS ~ Theorem – If an angle of one triangle is congruent to an angle of a second triangle, and the sides including the two angles are proportional, then the triangles are similar.
A' C'
B'
A
B
C
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Similar Triangles (7-3)Similar Triangles (7-3)
SSS ~ Theorem – If the corresponding sides of two triangles are proportional, then the triangles are similar.
SSS ~ Theorem – If the corresponding sides of two triangles are proportional, then the triangles are similar.
A' C'
B'
A
B
C
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Similar Triangles (7-3)Similar Triangles (7-3)
8
6
12
9
I
E
F
G
H
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Similar Triangles (7-3)Similar Triangles (7-3)
Are the triangles similar? If so, write a similarity statement and name the postulate or theorem you used. If not, explain.
No the vertical angle is not between the two pairs of proportional sides.
Are the triangles similar? If so, write a similarity statement and name the postulate or theorem you used. If not, explain.
No the vertical angle is not between the two pairs of proportional sides.
86
12
9
I
E
F
G
H
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Similar Triangles (7-3)Similar Triangles (7-3)
O
J K
L M
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Similar Triangles (7-3)Similar Triangles (7-3)
Find the value of x. Find the value of x.
x
25
40
10R S
T
U V
W
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Indirect Measurement (7-3)Indirect Measurement (7-3)
When a 6 ft man casts a shadow 18 ft long, a nearby tree casts a shadow 93 ft long. How tall is the tree?
When a 6 ft man casts a shadow 18 ft long, a nearby tree casts a shadow 93 ft long. How tall is the tree?
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HomeworkHomework
7-4 A Postulate for Similar Triangles (AA) worksheet #1-12 all
7-5 Theorems For Similar Triangles (SSS and SAS) worksheet #1-6 all
Similar Triangles Worksheet (all three methods)
7-4 A Postulate for Similar Triangles (AA) worksheet #1-12 all
7-5 Theorems For Similar Triangles (SSS and SAS) worksheet #1-6 all
Similar Triangles Worksheet (all three methods)
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Similarity in Right Triangles (7-4)Similarity in Right Triangles (7-4)
Theorem: The altitude to the hypotenuse of a right triangle divides the triangle into two triangles that are similar to the original triangle and to each other.
Theorem: The altitude to the hypotenuse of a right triangle divides the triangle into two triangles that are similar to the original triangle and to each other.
DA
B
C
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Similarity in Right Triangles (7-4)Similarity in Right Triangles (7-4)
Geometric mean
For any two positive numbers a and b, x is the geometric mean if
Another way to find the geometric mean:
Geometric mean
For any two positive numbers a and b, x is the geometric mean if
Another way to find the geometric mean:
abx
b
x
x
a
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Similarity in Right Triangles (7-4)Similarity in Right Triangles (7-4)
Find the geometric mean of 32 and 2.
Find the geometric mean of 6 and 20.
Find the geometric mean of 32 and 2.
Find the geometric mean of 6 and 20.
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Similarity in Right Triangles (7-4)Similarity in Right Triangles (7-4)
2
6
xQ N
O
P
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Similarity in Right Triangles (7-4)Similarity in Right Triangles (7-4)
5
x
4 BY
Z
A
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Similarity in Right Triangles (7-4)Similarity in Right Triangles (7-4)
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HomeworkHomework
8-1 worksheet #24-31 all 8-1 worksheet #24-31 all
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Proportions in Triangles (7-5)Proportions in Triangles (7-5)
Side-Splitter Theorem – If a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally.
Side-Splitter Theorem – If a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally.
y
6
10
12E
A
B
C
D
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Proportions in Triangles (7-5)Proportions in Triangles (7-5)
Solve for x.
x = 9
Solve for x.
x = 9
x + 1
x + 6
x - 3xO
K
L
M
N
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Proportions in Triangles (7-5)Proportions in Triangles (7-5)
Corollary: If three parallel lines intersect two transversals, then the segments intercepted on the transversals are proportional.
Corollary: If three parallel lines intersect two transversals, then the segments intercepted on the transversals are proportional.
b d
ca
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Proportions in Triangles (7-5)Proportions in Triangles (7-5)
Solve for x.
x = 24
Solve for x.
x = 24
21
14
x
16
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Proportions in Triangles (7-5)Proportions in Triangles (7-5)
Triangle-Angle-Bisector Theorem – If a ray bisects an angle of a triangle, then it divides the opposite side into two segments that are proportional to the other two sides of the triangle.
x = 18
Triangle-Angle-Bisector Theorem – If a ray bisects an angle of a triangle, then it divides the opposite side into two segments that are proportional to the other two sides of the triangle.
x = 18
30
x
24
40
J
G H
I
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Proportions in Triangles (7-5)Proportions in Triangles (7-5)
Solve for x.
x = 11.25
Solve for x.
x = 11.25
12.5
9x
10
NK
L
M
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HomeworkHomework
7-6 Proportional Lengths worksheet Proportional Parts in Triangles and Parallel
Lines worksheet
p.475 #9-12, 15-22 Study for test
7-6 Proportional Lengths worksheet Proportional Parts in Triangles and Parallel
Lines worksheet
p.475 #9-12, 15-22 Study for test