Transcript Similar Triangles I

Course 3 5-5
Class Notes
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In geometry, two polygons
are similar when one is a
replica (scale model) of
the other.
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Consider Dr. Evil and Mini Me from
Mike Meyers’ hit movie Austin Powers.
Mini Me is supposed to be an exact
replica of Dr. Evil.
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The following are similar figures.
I
II
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The following are non-similar figures.
I
II
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Feefee the mother cat, lost her daughters, would you
please help her to find her daughters. Her daughters have
the similar footprint with their mother.
Feefee’s
footprint
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1.
Which of the following is similar to the
above triangle?
A
B
C
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Similar triangles are triangles that have the same
shape but not necessarily the same size.
A
D
E
B
F
C
ABC  DEF
When we say that triangles are similar there are several
repercussions that come from it.
A  D
B  E
C  F
AB
DE
=
BC
EF
=
AC
DF
Six of those statements are true as a result of the
similarity of the two triangles. However, if we need to
prove that a pair of triangles are similar how many of
those statements do we need? Because we are working
with triangles and the measure of the angles and sides
are dependent on each other. We do not need all six.
There are three special combinations that we can use
to prove similarity of triangles.
1. SSS Similarity Theorem
 3 pairs of proportional sides
2. SAS Similarity Theorem
 2 pairs of proportional sides and congruent
angle between them
3. AA Similarity Theorem
 2 pairs of congruent angles
Side-Side-Side Similarity (SSS ~ )
Theorem
• If the corresponding sides of two triangles are
proportional, then the triangles are similar.
E
1. SSS Similarity Theorem
 3 pairs of proportional sides
A
9.6
5
B
C
12
mAB

mDF
mBC

mFE
5
 1.25
4
12
 1.25
9.6
F
4
mAC
13

 1.25
mDE 10.4
ABC  DFE
D
Side-Angle-Side Similarity (SAS ~)
Theorem
• If an angle of one triangle is congruent to an angle of
a second triangle, and the sides that include the two
angles are proportional, then the triangles are similar.
2. SAS Similarity Theorem
 2 pairs of proportional sides and congruent
angles between them
L
G
70
H
7
I
mGH
5

 0.66
7 .5
mLK
mHI
7

 0.66
mKJ 10.5
70
J
10.5
mH = mK
GHI  LKJ
K
The SAS Similarity Theorem does not work unless
the congruent angles fall between the proportional
sides. For example, if we have the situation that is
shown in the diagram below, we cannot state that the
triangles are similar. We do not have the information
that we need.
L
G
50
H
7
I
J
50
K
10.5
Angles I and J do not fall in between sides GH and HI and
sides LK and KJ respectively.
7.3 Proving Triangles Similar
• Angle-Angle Similarity (AA ~) Postulate
– If two angles of one triangle are congruent to
two angles of another triangle, then the
triangles are similar.
3. AA Similarity Theorem
 2 pairs of congruent angles
Q
M
70
50
N
mN = mR
mO = mP
O
50
70
P
MNO  QRP
R
It is possible for two triangles to be similar when
they have 2 pairs of angles given but only one of
those given pairs are congruent.
T
X
Y
34
34
59
59
Z
87 59
U
S
mS = 180- (34 + 87)
mS = 180- 121
mS = 59
mT = mX
mS = mZ
TSU  XZY
Note: One triangle is a scale model of the
other triangle.
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How do we know if two
triangles are similar or
proportional?
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Triangles are similar (~)
if corresponding angles
are equal and the ratios
of the lengths of
corresponding sides are
equal.
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Interior Angles of Triangles
B
A
C
The sum of the measure
of the angles of a
triangle is 1800.
C 1800
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Determine whether the pair of triangles is similar.
Justify your answer.
Answer: Since the corresponding angles have equal
measures, the triangles are similar.
If the product of the
extremes equals the
product of the means
then a proportion
exists.
a c

b d
bc  ad
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This
 ABC and AC
 XYZ are
ABtells us thatBC
=K
=K
=K
similar and proportional.
YZ
XZ
XY
12
2
6
8
2
4
10
2
5
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Q: Can these triangles be similar?
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Answer—Yes, right triangles can also
be similar but use the criteria.
AB
=
XY
AC
BC
=K
=
XZ
YZ
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AB
=
XY
BC
=
YZ
AC
=K
XZ
6
8
10
=
=
=K
4
6
8
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Do we have equality?
6
8
10
=
=
=K
4
6
8
6
8
= 1.5 but
= 1.3
4
6
This tells us our triangles are not
similar. You can’t have two different
scaling factors!
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If we are given that two
triangles are similar or
proportional what can we
determine about the
triangles?
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The two triangles below are known to
be similar, determine the missing value
X.
7.5 4.5

5
x
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7.5 4.5

5
x
54.5  7.5x
22.5  7.5x
3 x
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In the figure, the two triangles are similar.
What are c and d ?
A
5
P
10
c
B
R
4
d
Q
6
10 c

5 4
C
40  5c
8c
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In the figure, the two triangles are similar.
What are c and d ?
A
5
P
10
c
B
R
4
d
Q
6
C
10 6

30  10d
5 d
3d
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Sometimes we need to measure a distance
indirectly. A common method of indirect
measurement is the use of similar triangles.
17 6

102 h
36  h
h
6
17
102
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Algebra 1 Honors 4-2
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