Download Geometry_Grade 912 Similarity and Congruence

Similarity and Congruence
Similarity and
Congruence
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Similarity
and Congruence
SIMILARITY
AND CONGRUENCE
If two shapes are congruent, it means thay are equal in every way – all their corresponding sides and
angles are equal. Similar figures have the same shape, but not necessarily the same size. In this book,
it is shown how similar and congruent shapes can be useful in solving problems.
Try to answer these questions now, before working through the chapter.
I used to think:
The symbol for congruent is /. What do you think it means to say TABC / TDEF?
If a square with side length 4cm has been enlarged by a scale factor of 2, then what is the side length of the
large square?
If two triangles are the same except for one angle, are they congruent?
Answer these questions, after working through the chapter.
But now I think:
The symbol for congruent is /. What do you think it means to say TABC / TDEF?
If a square with side length 4cm has been enlarged by a scale factor of 2, then what is the side length of the
large square?
If two triangles are the same except for one angle, are they congruent?
What do I know now that I didn’t know before?
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Similarity and Congruence
Basics
Congruent Triangles (/)
Congruent triangles are shapes that are exactly the same in every way (side lengths and interior angles
are all equal). If even one side or one angle are not equal, then the triangles are not congruent.
Congruent Triangles
These triangles are not congruent
B
11.4
A
60c
7.4
40c
80c
10
10
80c
D
7.4
40c
Q
C
E
24.5
R
60c
60c
22.6
22.6
20
P
11.4
M
50c
70c
N
F
Angles: +A = +P
+B = +Q
+C = +R
Sides: AB = PQ
BC = QR
CA = RP
48c
P
Angle is different
All sides and angles are equal
therefore triangles are congruent.
No angle in TDEF is equal to +N .
` These triangles are NOT congruent.
From above, ∆ABC and ∆PQR are congruent. Using the proper notation, this is written as ∆ABC / ∆PQR. It is
important to make sure the angles match when using the / symbol. Here is an example:
Show that these triangles are congruent
A
M
75c
10
B
75c
14
65c
10
40c
15
C
N
65c
40c
15
In ∆ABC and ∆MNP: +A = +M = 75c
+B = +N = 65c
14
P
AB = MN
angles
BC = NP
+C = +P = 40c
sides
AC = MP
` ∆ABC / ∆MNP
Notice the order of the letters when using /. The equal angles are written in the same order.
Equal angles written in the
same order (correct):
Equal angles not written in
the same order (incorrect):
TABC / TMNP
TABC / TPMN
TBCA / TNPM
TCAB / TPMN
2
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
TCAB / TPNM
TBCA / TMNP
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
Similarity and Congruence
Basics
Similar Figures ( ||| )
Similar figures have the same shape, but not necessarily the same size. These shapes are similar.
K
10 cm
20 cm
B
J
A
5 cm
10 cm
9 cm
D
18 cm
M
15 cm
C
30 cm
L
Similar figures have two important properties:
•
•
Their corresponding angles are equal.
Their corresponding sides are in the same ratio. In the above similar shapes, the ratio of the corresponding sides is 2 since the sides in the bigger shape are double the length in the smaller shape.
These shapes are similar
P
8 cm
A
Q
24 cm
B
45c
3 cm
S
R
D
a
Find the length of AD.
b
18 cm
135c
C
Find the size of +B .
PQRS and ABCD are similar.
PQRS and ABCD are similar.
` AD =
PS
AD
`
=
3
` +B = +Q
AB
PQ
24
8
` +B = 45c
` AD = 9cm
c
Find the size of +R .
d
Find the size of RS.
PQRS and ABCD are similar.
PQRS and ABCD are similar.
` +R = +C
` RS =
DC
` RS =
18
` +R = 135c
PQ
AB
8
24
` RS = 6cm
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Questions
Basics
1. Show these triangles are congruent, and then use / symbol to state congruency.
T
a
A
b
F
D
10
14
10
8
E
14
V
8
c
G
U
A
13
B
67c
13
C
10
P
d
D
13
67c
E
10
12
M
C
25c
B
F
25c
E
12
23c
67c
R
4
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67c
5
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Q
L
F
K
23c
95c
60c
67c
13
Similarity and Congruence
Questions
Basics
2. Find the missing values in these similar shapes (all measurements in cm):
a
P
D
85c
7
b
110c E
E
14
Q
G 95c
70c
110c
T
9
130c
Q
S
S
D
F
3
150c B
12
9
10
P
A
6
6
100c
50c
15
R
18
R
PQ =
QR =
PT =
ST =
RS =
+P =
+D =
+P =
+S =
+Q =
QR =
+C =
L
c
J
C
d
C
4 K
40c
E
F
M
N
B
A
H
8
15
M
5
12
G
L
45c
K
D
LM =
AB =
+M =
+A =
Given LM = 2 . Find the length of KN.
EF
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Knowing More
Testing for Congruent Triangles
Congruent triangles have all 3 corresponding sides equal, and all 3 corresponding angles equal – that is 6 properties.
However, there are tests for congruent triangles that don’t require showing all 6 properties. There are four tests:
Side Side Side (SSS)
Side Angle Side (SAS)
If the corresponding sides of two
triangles are equal, then the triangles are
congruent (SSS).
If 2 sides and the included angle are
respectively equal, then the triangles are
congruent (SAS).
Q
B
A
C
P
R
A
C
L
N
In ∆ABC and ∆PQR:
In ∆ABC and ∆LMN:
AB = PQ
AB = LM
BC = QR
+A = +L
AC = PR
AC = LN
TABC / TPQR
TABC / TLMN (SAS)
(SSS)
Side Angle Angle (SAA)
Right Angle, Hypotenuse, Side (RHS)
If 2 corresponding sides and a corresponding
angle are equal, then the triangles are
congruent (SAA).
If two right angled triangles have the same
hypotenuse, and a corresponding side, then the
triangles are congruent (RHS).
K
A
A
N
L
B
M
C
C
In ∆ABC and ∆KLM:
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B
M
O
In ∆ABC and ∆NOM:
AB = KL
AC = NM
+A = +K
AB = NO
+C = +M
+C = +M = 90c
TABC / TKLM
6
M
B
TABC / TNOM (RHS)
(SAA)
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Similarity and Congruence
Knowing More
Here are some examples:
Show that these triangles are congruent:
a
L
In TIJK
I
+J = 180c - 93c - 62c
62c
93c M
25c
12 cm
= 25c
K 93c
` +N = +J
12 cm
N
J
b
(Both are 25c )
In TLMN and TIJK :
+N = +J
(Proved above)
+M = +K = 93c
(Given)
MN = KJ = 12cm
(Given)
` TLMN / TIKJ
(SAA)
In TDEF and TGEF :
E
DF = GF
(Given)
DE = GE
(Given)
EF is common
G
D
(Angle sum of triangle)
` TDEF / TGEF
F
(SSS)
Here is an example where congruence is used to show something is true.
Show that BD bisects + ABC in the diagram below
In TABD and TCBD :
B
AB = BC
(Given)
BD is common
A
D
C
+ADB = +CDB = 90c
(Given)
` +ABD / +CBD
(RHS)
` +ABD = +CBD
(Congruent triangles, TABD / TCBD )
` BD bisecting +ABC
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Knowing More
( ||| )
Similar Triangles
There are two ways to show that triangles are similar:
•
•
Show that their corresponding sides are in proportion.
Show that they have equal angles (AAA).
If two triangles are similar, the symbol ||| is used.
Show that these triangles are similar:
a
In ∆ABC:
A
78c
+ C = 180c - 58c - 78c = 44c
(Angle sum of a triangle)
In ∆GHI:
58c
B
C
H
(Angle sum of a triangle)
In ∆ABC and ∆GHI:
44c
G
+ G = 180c - 58c - 44c = 78c
+C = +H
(Both are 44c )
58c
+A = +G
(Both are 78c )
I
+B = +I
(Both are 58c )
` ∆ABC ||| ∆GIH
(AAA)
b
R
In ∆QRS and ∆TUV:
12
Q
TU = 18 = 3
RQ
12
2
22
UV = 33 = 3
RS
22
2
18
S
TV = 27 = 3
QS
18
2
U
` All sides in proportion
18
` ∆QRS ||| ∆TUV
33
T
27
V
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c
TU = UV = TV
m
RQ
RS
QS
Similarity and Congruence
Questions
Knowing More
1. Explain what the following mean:
a
/
b
SSS
c
SAS
d
SAA
e
RHS
f
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Questions
2. Prove these triangles are congruent:
a
C
E
5
5
B
4
A
F
b
D
3
B
A
C
E
D
c
P
M
10
S
75c
N
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75c
75c
R
Q
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Knowing More
Similarity and Congruence
Questions
Knowing More
3. In the diagram below, show that ∆BCD ||| ∆ACE.
C
B
D
A
E
4. Prove that ∆JKL ||| ∆STU.
J
T
8
L
15
6
K
S
12
U
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Similarity and Congruence
Using Our Knowledge
Scale Factor in Similar Triangles
When triangles are similar, their angles are equal (AAA) and their corresponding sides are in proportion. The ratio
that their sides are in proportion is called the Scale Factor. A Scale Factor either enlarges (scales up) or reduces
(scales down).
G
A
D
2
E
F
4
8
4
1
Scale factor =
2
3
12
Scale factor =2
6
B
C
8
H
16
In ∆ABC and ∆DEF:
In ∆ABC and ∆GHI:
DF = 3 = 1
AC
6
2
GI = 12 = 2
AC
6
scale factor of
∆DEF to ∆ABC
EF = 4 = 1
BC
8
2
HI = 16 = 2
BC
8
DE = 2 = 1
AB
4
2
GH = 8 = 2
AB
4
` TABC ||| TDEF
` TABC ||| TGHI
scale factor of
∆GHI to ∆ABC
If the scale factor is bigger than 1, the triangle is enlarged. If the scale factor is between 0 and 1 (decimal or fraction),
the triangle is reduced.
Show these triangles are similar and find their scale factor of ∆PQR to ∆LMN
L
In ∆LMN and ∆PQR:
6cm
PQ
= 21 = 3
LM
7
N
7cm
3cm
QR
= 9 =3
MN
3
M
P
18cm
` ∆LMN ||| ∆PQR
R
21cm
9cm
RP = 18 = 3
NL
6
(Corresponding sides are in proportion)
PQ
QR
=
= RP = 3
LM
MN
NL
` The scale factor of ∆PQR to ∆LMN is 3.
Q
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Similarity and Congruence
Using Our Knowledge
Using Similar Triangles
If triangles are known to be similar, then the properties of similar triangles can be used to solve problems.
Find the values of x and y if ∆ABC ||| ∆TUV (all measurements in cm)
A
In ∆ABC and ∆TUV:
x
To find x:
10
C
AC = AB
TV
TU
5
B
(Similar triangles, ∆ABC ||| ∆TUV)
` x = 10
18
30
T
` x = 18 # 10
30
18
` x = 6cm
30
V
To find y:
y
U
UV = TU
BC
AB
`
(Similar triangles, ∆ABC ||| ∆TUV)
y
= 30
5
10
` y = 5 # 30
10
` y = 15cm
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Questions
Using Our Knowledge
1. Find the scale factor in these pairs of similar triangles for both the smaller and larger triangles:
Given ∆RST ||| ∆UVW.
a
R
35
S
50
25
T
U
7
V
5
10
W
Given ∆ABE ||| ∆ACD.
b
C
4
B
8
6
A
14
E
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D
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Similarity and Congruence
Questions
Using Our Knowledge
2. Answer these questions about the diagram below:
J
10
N
12
8
K
5
M
a
Show that ∆JML ||| ∆JNK.
b
Find the length of KL.
c
Find the length of ML.
L
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Questions
Using Our Knowledge
3. Answer these questions about the shape below:
F
6
8
H
25
G
J
10
I
a
Show that ∆GFH ||| ∆GIJ.
b
Find the length of GI.
c
Find the length of IJ.
d
Find the scale factor of the larger triangle with respect to the small triangle.
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Similarity and Congruence
Thinking More
Using Congruence and Similarity in Proofs
Congruence and similarity are used to prove properties of triangles, quadrilaterals and other shapes.
Show that the diagonals of a parallelogram bisect each other
A
B
Draw in diagonals (AC and BD) in the parallelogram ABCD:
Given:
O
AB || CD
AB = CD
D
AD || BC
C
AD = BC
To prove: AO = OC and BO = OD
Proof:
In ∆AOB and ∆COD
AB || CD
(Given)
` +CAB = +DCA
(Alternate angles; AB || CD)
and +ABD = +BDC
(Alternate angles; AB || CD)
AB = CD
(Given)
` TAOB / TCOD
(SAA)
` AO = OC and BO = OD
(Congruent triangles; TAOB / TCOD )
` The diagonals of a parallelogram bisect each other.
Similarity can also be used in proofs.
In the diagram below, prove that BE || CD if AB = AE
AC
AD
A
Given:
To prove: BE || CD
E
B
C
AB = AE
AC
AD
D
Proof:
In ∆ABE and ∆ACD
AB = AE
AC
AD
(Given)
` TABE / TACD
(Corresponding sides in proportion)
` ACD / TABE
(Similar triangles; TABE ||| TACD )
` BE || CD
(Given)
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Questions
Thinking More
1. Answer these questions about PQRS below given that PQ = RS and PR = QS:
P
Q
R
S
a
Prove ∆PRS / ∆SQP.
b
Prove PQRS is a parallelogram (PQ || RS and PS || QR).
c
Prove that the opposite angles of a parallelogram are equal.
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Similarity and Congruence
Questions
Thinking More
2. ∆KLM is an iscosceles triangle with KL = KM .
K
M
L
N
a
KN has been drawn to bisect ML. Show that ∆KMN / ∆KLN.
b
Show that + MNK = + LNK = 90c.
c
Prove that + M = + L.
3. In ∆DEF, + F = + E, if the line GD bisects + FDE. Prove ∆DEF is isosceles.
E
G
F
D
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Questions
4. Prove the following about the Rhombus STUV below:
S
T
O
V
U
a
∆VOS / ∆TOU
b
∆SOT / ∆UOV
c
Show that the diagonals bisect each other.
d
Show that the diagonals bisect at 90c.
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Thinking More
Similarity and Congruence
Answers
Basics:
2. a
b
c
d
Knowing More:
PQ = 12cm
QR = 18cm
RS = 20cm
+P = 85c
+S = 95c
+Q = 110c
PT = 2cm
ST = 4cm
+D = 100c
+P = 110c
QR = 5cm
+C = 50c
LM = 10cm
AB = 2 2 cm
3
+M = 45c
+A = 40c
1. e RHS means Right Angle, Hypotenuse, Side.
It is one of the four tests that can be used
to prove two triangles are congruent.
If two right angle triangles have equal
hypotenuse and an equal corresponding
side, then the triangles are congruent.
f
means is similar to. It is used to
show two triangles have the same shape
(corresponding angles are equal and
corresponding sides are in proportion).
Using Our Knowledge:
1. a Scale factor from ∆RST to ∆UVW is 1
5
Scale factor from ∆UVW to ∆RST is 5
KN = 10cm
b
Knowing More:
1. a / symbol means is congruent to. It is used
to show two triangles are exactly the same
in every way (corresponding sides equal
and corresponding angles equal).
b
||| symbol
SSS means Side, Side, Side. It is one of
the four tests that can be used to prove
two triangles are congruent. If the
corresponding sides of two triangles are
equal, then the triangles are congruent.
c
SAS means Side, Angle, Side. It is one of
the four tests that can be used to prove
two triangles are congruent. If two sides
and the included angle of two triangles are
equal, then the triangles are congruent.
d
SAA means Side, Angle, Angle. It is one
of the four tests that can be used to
prove two triangles are congruent. If a
corresponding side and two corresponding
angles of two triangles are equal, then the
triangles are congruent.
2. b KL = 6cm
c
ML = 12cm
3. b GI = 15cm
c
IJ = 20cm
d
Scale factor from ∆GFH to ∆GIJ is 2 1
2
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Scale factor from ∆ABE to ∆ACD is 1 1
2
Scale factor from ∆ACD to ∆ABE is 2
3
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Notes
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