Download Similarity - MrsMcFadin

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
 Examples: 2 squares that have different
lengths of sides.
 2 regular hexagons
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
m F = ?
m C = ?
Similar Polygons (7-2)
AB
ABCD ~ EFGH
Complete the statements.
EF
mF = ?
mC = ?
HG
C
D
DC
=
=
BC
?
FG
?
G
H
115
65
B
F
A
E
Similar Polygons (7-2)
 Determine whether the parallelograms are
similar. Explain.
1
2
114
2
2
4
4
59
1
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
Similar Polygons (7-2)
 If ABC ~ YXZ, find the scale factor of
the large triangle to the small and find the
Y
value of x.
A
40
x
B
12
C
Z
X
scale factor = 5/2 x= 16
30
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)
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
Similar Polygons (7-2)
 Homework
 Similar Polygons worksheet #1-17
odd
 Using Similar Polygons worksheet
#1-15 odd
Scale Drawing
 Problem 2 on p.443
 Complete Similarity Application Problems
 More practice
p.444 #9, 13, 15, 19, 23, and 25
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.
B'
B
A
C
A'
C'
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
B'
are similar.
B
A
C
A'
C'
Similar Triangles (7-3)
 SSS ~ Theorem – If the corresponding sides
of two triangles are proportional, then the
B'
triangles are similar.
B
A
C
A'
C'
Similar Triangles (7-3)
F
H
6
9
I
8
E
12
G
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.
F
H
6
9
8
I
E
12
 No the vertical angle is not between the two pairs of
proportional sides.
G
Similar Triangles (7-3)
J
K
O
L
M
Similar Triangles (7-3)
 Find the value of x.
R
10
W
S
x
40
T
U
25
V
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?
Homework
 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)
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.
B
A
D
C
Similarity in Right Triangles (7-4)
 Geometric mean
For any two positive numbers a and b, x is
the geometric mean if
a x
x

b
Another way to find the geometric mean:
x  ab
Similarity in Right Triangles (7-4)
 Find the geometric mean of 32 and 2.
 Find the geometric mean of 6 and 20.
Similarity in Right Triangles (7-4)
O
6
P
2
Q
x
N
Similarity in Right Triangles (7-4)
Z
x
Y
4
B
5
A
Similarity in Right Triangles (7-4)
Homework
 8-1 worksheet #24-31 all
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.
B
y
D
6
12
E
A
10
C
Proportions in Triangles (7-5)
 Solve for x.
L
x
x-3
O
N
x+1
x+6
M
K
 x=9
Proportions in Triangles (7-5)
 Corollary: If three parallel lines intersect
two transversals, then the segments
intercepted on the transversals are
proportional.
a
b
c
d
Proportions in Triangles (7-5)
 Solve for x.
14
16
x
 x = 24
21
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
I
triangle.
x
J
24
30
 x = 18
G
40
H
Proportions in Triangles (7-5)
 Solve for x.
9
M
N
10
K
12.5
L
 x = 11.25
x
Homework
 7-6 Proportional Lengths worksheet
 Proportional Parts in Triangles and Parallel
Lines worksheet
 p.475 #9-12, 15-22
 Study for test