asdSimilar triangles 5 5 (1)

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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A B C
1.
Which of the following is similar to the
above triangle?

Similar triangles are triangles that have the same
shape but not necessarily the same size.
A
C
B
D
F
E
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= =

1. PPP Similarity Theorem
 3 pairs of proportional sides
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.
2. PAP Similarity Theorem
 2 pairs of proportional sides and congruent
angles between them
3. AA Similarity Theorem
 2 pairs of congruent angles

1. PPP Similarity Theorem
 3 pairs of proportional sidesA
B C
E
F D
251
4
5
.
DFm
ABm

251
69
12
.
.FEm
BCm

251
410
13
.
.DEm
ACm

5
4
12
9.6
ABC  DFE

2. PAP Similarity Theorem
 2 pairs of proportional sides and congruent
angles between them
G
H I
L
J K
660
57
5
.
.LKm
GHm

660
510
7
.
.KJm
HIm

7
10.5
70
70
mH = mK
GHI  LKJ

The PAP 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.
G
H I
L
J K
7
10.5
50
50
Angles I and J do not fall in between sides GH and HI and
sides LK and KJ respectively.

3. AA Similarity Theorem
 2 pairs of congruent angles
M
N O
Q
P R
70
70
50
50
mN = mR
mO = mP MNO  QRP

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.
87
34
34
S
T
U
XY
Z
mT = mX
mS = 180- (34 + 87)
mS = 180- 121
mS = 59
mS = mZ
TSU  XZY
59
5959
34
34

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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A
B
C
The sum of the measure
of the angles of a
triangle is 1800.
C 1800
Interior Angles of Triangles

Determine whether the pair of triangles is similar.
Justify your answer.
Answer: Since the corresponding angles have equal
measures, the triangles are similar.

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If the product of the
extremes equals the
product of the means
then a proportion
exists.
d
c
b
a

adbc 

AB
=K
XY
BC
=K
YZ
AC
=K
XZ
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2
6
12
 2
4
8
 2
5
10

This tells us that  ABC and  XYZ are
similar and proportional.

Q: Can these triangles be similar?
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Answer—Yes, right triangles can also
be similar but use the criteria.
AB
=
XY
BC
=
YZ
AC
= K
XZ
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6 8 10
= = = K
4 6 8
AB
=
XY
BC
=
YZ
AC
= K
XZ
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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!
Do we have equality?
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If we are given that two
triangles are similar or
proportional what can we
determine about the
triangles?

The two triangles below are known to
be similar, determine the missing value
X.
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x
5.4
5
5.7


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x
5.4
5
5.7

  x5.75.45 
x5.75.22 
x3

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A
B
C
P
Q
R10
6
c
5
4 d
In the figure, the two triangles are similar.
What are c and d ?
45
10 c
 c540  c8

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A
B
C
P
Q
R10
6
c
5
4 d
In the figure, the two triangles are similar.
What are c and d ?
d
6
5
10
 d1030  d3

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Sometimes we need to measure a distance
indirectly. A common method of indirect
measurement is the use of similar triangles.
h
6
17
102
h
6
102
17

h36

Algebra 1 Honors 4-2
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asdSimilar triangles 5 5 (1)

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