www.mathletics.com
Similarity and Congruence
Curriculum Ready
Similarity andCongruence
ACMMG: 201, 220, 221, 243, 244
SERIES TOPIC
J 1100% Similarity and Congruence
Mathletics 100% © 3P Learning
Similarity and Congruence
10
SIMILARITY AND CONGRUENCE
Try to answer these questions now, before working through the chapter.
Answer these questions, after working through the chapter.
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.
But now I think:
What do I know now that I didn’t know before?
I used to think:
The symbol for congruent is /. What do you think it means to say TABC / TDEF?
The symbol for congruent is /. What do you think it means to say TABC / TDEF?
If two triangles are the same except for one angle, are they congruent?
If two triangles are the same except for one angle, are they congruent?
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 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?
SERIES TOPIC
J 102 100% Similarity and Congruence
Mathletics 100% © 3P Learning
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
All sides and angles are equal therefore triangles are congruent.
In ∆ABC and ∆MNP:
angles
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:
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):
No angle in DEFT is equal to N+ .` These triangles are NOT congruent.
Show that these triangles are congruent
B
11.4
7.4
10A
C
80c
60c
40c R
11.4
7.410
PQ
80c
60c40c
Angles: A P
B Q
C R
+ +
+ +
+ +
=
=
=
Sides: AB PQ
BC QR
CA RP
=
=
=
24.5
22.6
20
E
D
F
50c
70c
60c22.6
M
N P48c
Angle is different
10 14
15B C
A
75c
65c 40c
10 14
15N P
M
75c
65c 40c
A M
B N
C P
75
65
40
+ +
+ +
+ +
= =
= =
= =
c
c
c
sides
AB MN
BC NP
AC MP
=
=
=
ABC MNP
BCA NPM
CAB PMN
T T
T T
T T
/
/
/
ABC PMN
CAB PNM
BCA MNP
T T
T T
T T
/
/
/
` ∆ABC / ∆MNP
SERIES TOPIC
J 3100% Similarity and Congruence
Mathletics 100% © 3P Learning
Similarity and Congruence
10
Basics
Similar Figures |||( )
Similar figures have the same shape, but not necessarily the same size. These shapes are similar.
Find the length of AD.
Find the size of R+ . Find the size of RS.
Find the size of B+ .
PQRS and ABCD are similar.
PQRS and ABCD are similar. PQRS and ABCD are similar.
PQRS and ABCD are similar.
These shapes are similar
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.
a
c d
b
D
C
10 cm
15 cm
5 cm 9 cm
K
L
M
J
10 cm
20 cm
18 cm
30 cm
A
B
P
S R
Q
3 cm
8 cm
45c
A
D C
B24 cm
18 cm135c
cm
PSAD
PQAB
AD
AD
3 824
9
`
`
`
=
=
=
B Q
B 45
`
`
+ +
+
=
= c
R C
R 135
`
`
+ +
+
=
= c
cm
DCRS
ABPQ
RS
RS
18 248
6
`
`
`
=
=
=
SERIES TOPIC
J 104 100% Similarity and Congruence
Mathletics 100% © 3P Learning
Similarity and Congruence BasicsQuestions
1. Show these triangles are congruent, and then use / symbol to state congruency.
10
8
14
V U
T
10
8
14
E
F
G
13
10
13
A
B C67c
10
13
D
E F67c 67c
A
C B60c 25c
D
F E
95c
25c
a
c
b
d
12 13
5
P
QR
23c
67c
5
12
13
M
L
K
23c
67c
SERIES TOPIC
J 5100% Similarity and Congruence
Mathletics 100% © 3P Learning
Similarity and Congruence
10
Questions Basics
2. Find the missing values in these similar shapes (all measurements in cm):
Given EFLM 2= . Find the length of KN.
a
PQ = QR =
RS = P+ =
S+ = Q+ =
PT =
D+ =
QR =
ST =
P+ =
C+ =
LM = AB =
M+ = A+ =
c d
b
S
P
Q
R
14
D
E
F
G
76
9
10
110c
70c
95c
85c T
P
Q
R
S100c
50c
E
A
B
C
D
6
9 3
15
18
12
110c
130c
150c
L
K
M
J4
8
40cA
B
C
D
12
15
45c
E F
GH5
K
L
M
N
SERIES TOPIC
J 106 100% Similarity and Congruence
Mathletics 100% © 3P Learning
Similarity and Congruence Knowing More
Testing for Congruent Triangles
Side Side Side (SSS)
If the corresponding sides of two triangles are equal, then the triangles are
congruent (SSS).
Side Angle Angle (SAA)
If 2 corresponding sides and a corresponding angle are equal, then the triangles are
congruent (SAA).
Side Angle Side (SAS)
If 2 sides and the included angle are respectively equal, then the triangles are
congruent (SAS).
Right Angle, Hypotenuse, Side (RHS)
If two right angled triangles have the same hypotenuse, and a corresponding side, then the
triangles are congruent (RHS).
In ∆ABC and ∆PQR:
In ∆ABC and ∆KLM:
In ∆ABC and ∆LMN:
In ∆ABC and ∆NOM:
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:
AB PQ
BC QR
AC PR
=
=
=
AB KL
A K
C M
+ +
+ +
=
=
=
AC NM
AB NO
C M 90c+ +
=
=
= =
AB LM
A L
AC LN
+ +
=
=
=
ABC PQRT T/
ABC KLMT T/ ABC NOMT T/
ABC LMNT T/(SSS)
(SAA) (RHS)
(SAS)
B
CA
Q
P R
B
CA
M
L N
A
B
C
K
L
M
A
BC
N
OM
SERIES TOPIC
J 7100% Similarity and Congruence
Mathletics 100% © 3P Learning
Similarity and Congruence
10
Knowing More
Here are some examples:
Here is an example where congruence is used to show something is true.
Show that these triangles are congruent:
Show that BD bisects +ABC in the diagram below
a
b
L
N
M93c
12 cm 12 cm25c
62c
I
K
J
93c
In IJKT
In LMNT and IJKT :
In DEFT and GEFT :
In ABDT and CBDT :
180 93 62J
25
c c c
c
+ = - -
=
N J` + +=
cm
93
12
N J
M K
MN KJ
LMN IKJ`
c
+ +
+ +
T T/
=
= =
= =
is common
DF GF
DE GE
EF
DEF GEF` T T/
=
=
(Angle sum of triangle)
(Proved above)
(Given)
(Given)
(Given)
(SSS)
(Given)
(Given)
(Congruent triangles, ABD CBDT T/ )
(RHS)
(Given)
(SAA)
(Both are 25c )
D
E
F
G
B
DA C
is common
bisecting
90
AB BC
BD
ADB CDB
ABD CBD
ABD CBD
BD ABC
`
`
`
c+ +
+ +
+ +
+
/
=
= =
=
SERIES TOPIC
J 108 100% Similarity and Congruence
Mathletics 100% © 3P Learning
Similarity and Congruence Knowing More
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.
Similar Triangles |||( )
Show that these triangles are similar:
A
B C
H
G
I
58c
44c
Q
S
R
12
22
18
T
U
V
18
33
27
In ∆QRS and ∆TUV:
` All sides in proportion
` ∆QRS ||| ∆TUV
a
b
78c
58c
In ∆ABC:
+C = 180c - 58c - 78c = 44c
In ∆GHI:
+G = 180c - 58c - 44c = 78c
In ∆ABC and ∆GHI:
+C = +H
+A = +G
+B = + I
` ∆ABC ||| ∆GIH
(Angle sum of a triangle)
(Angle sum of a triangle)
(AAA)
(Both are 58c )
(Both are 78c )
(Both are 44c )
RQTU
RSUV
QSTV
1218
23
2233
23
1827
23
= =
= =
= =
RQTU
RSUV
QSTV= =c m
SERIES TOPIC
J 9100% Similarity and Congruence
Mathletics 100% © 3P Learning
Similarity and Congruence
10
Questions Knowing More
/
1. Explain what the following mean:
SSS
SAS
SAA
RHS
|||
a
b
c
d
e
f
SERIES TOPIC
J 1010 100% Similarity and Congruence
Mathletics 100% © 3P Learning
Similarity and Congruence Knowing MoreQuestions
c
2. Prove these triangles are congruent:
b
aC
AB
5
4
E
F D
5
3
A
B
C
D
E
M N
P
75c
S R
Q
75c 75c
SERIES TOPIC
J 11100% Similarity and Congruence
Mathletics 100% © 3P Learning
Similarity and Congruence
10
Questions Knowing More
3. In the diagram below, show that ∆BCD ||| ∆ACE.
4. Prove that ∆JKL ||| ∆STU.
A E
B D
C
J
KL
8
6
S
T
U
15
12
SERIES TOPIC
J 1012 100% Similarity and Congruence
Mathletics 100% © 3P Learning
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).
In ∆ABC and ∆DEF: In ∆ABC and ∆GHI:
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.
In ∆LMN and ∆PQR:
` ∆LMN ||| ∆PQR
` The scale factor of ∆PQR to ∆LMN is 3.
(Corresponding sides are in proportion)
Show these triangles are similar and find their scale factor of ∆PQR to ∆LMN
Q
R
P
9cm
18cm
21cm
N
L
M
3cm
6cm
7cm
B
A
C
46
8
D
EF
3
4
2
H
G
I
812
16
Scale factor =2Scale factor 21=
ACDF
BCEF
ABDE
63
21
84
21
42
21
= =
= =
= =
| | |ABC DEF` T T | | |ABC GHI` T T
scale factor of ∆DEF to ∆ABC
scale factor of ∆GHI to ∆ABC
ACGI
BCHI
ABGH
612 2
816 2
48 2
= =
= =
= =
LMPQ
MNQR
NLRP
721 3
39 3
618 3
= =
= =
= =
3LMPQ
MNQR
NLRP= = =
SERIES TOPIC
J 13100% Similarity and Congruence
Mathletics 100% © 3P Learning
Similarity and Congruence
10
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)
T
U
V
y
18
30
A
x
B
C
10
5
To find x:
To find y:
In ∆ABC and ∆TUV:
(Similar triangles, ∆ABC ||| ∆TUV)
(Similar triangles, ∆ABC ||| ∆TUV)
cm
TVAC
TUAB
x
x
x
18 3010
3018 10
6
`
` #
`
=
=
=
=
cm
10
BCUV
ABTU
y
y
y
530
105 30
15
`
` #
`
=
=
=
=
SERIES TOPIC
J 1014 100% Similarity and Congruence
Mathletics 100% © 3P Learning
Similarity and Congruence Using Our KnowledgeQuestions
1. Find the scale factor in these pairs of similar triangles for both the smaller and larger triangles:
Given ∆ABE ||| ∆ACD.b
a
R
S
T
50
35
25
Given ∆RST ||| ∆UVW.
UV
W
10
7
5
A
B
C
DE
8
4
63
SERIES TOPIC
J 15100% Similarity and Congruence
Mathletics 100% © 3P Learning
Similarity and Congruence
10
Questions Using Our Knowledge
2. Answer these questions about the diagram below:
a
b
c
Show that ∆JML ||| ∆JNK.
Find the length of KL.
Find the length of ML.
J
K
LM
N
10 12
85
SERIES TOPIC
J 1016 100% Similarity and Congruence
Mathletics 100% © 3P Learning
Similarity and Congruence Using Our KnowledgeQuestions
3. Answer these questions about the shape below:
a
b
c
d
Show that ∆GFH ||| ∆GIJ.
Find the length of GI.
Find the length of IJ.
Find the scale factor of the larger triangle with respect to the small triangle.
H
F
G
J
I
8 6
10
25
SERIES TOPIC
J 17100% Similarity and Congruence
Mathletics 100% © 3P Learning
Similarity and Congruence
10
Thinking More
Show that the diagonals of a parallelogram bisect each other
In the diagram below, prove that BE || CD if ACAB
ADAE=
Using Congruence and Similarity in Proofs
Congruence and similarity are used to prove properties of triangles, quadrilaterals and other shapes.
Similarity can also be used in proofs.
A
O
B
CD
Draw in diagonals (AC and BD) in the parallelogram ABCD:
Given:
Given:
| |
| |
AB CD
AB CD
AD BC
AD BC
=
=
| |BE CD
| |AB CD
To prove:
To prove:
AO = OC and BO = OD
Proof:
Proof:
In ∆AOB and ∆COD
and
CAB DCA
ABD BDC
` + +
+ +
=
=
and
AOB COD
AO OC BO OD
`
`
T T/
= =
AB = CD
` The diagonals of a parallelogram bisect each other.
(Given)
(Alternate angles; AB || CD)
(Alternate angles; AB || CD)
(Congruent triangles; AOB CODT T/ )
(SAA)
(Given)
A
D
E
C
B
ACAB
ADAE=
In ∆ABE and ∆ACD
ACAB
ADAE
ABE ACD
ACD ABE
`
`
T T
T
/
/
=
||BE CD`
(Given)
(Corresponding sides in proportion)
(Similar triangles; | | |ABE ACDT T )
(Given)
SERIES TOPIC
J 1018 100% Similarity and Congruence
Mathletics 100% © 3P Learning
Similarity and Congruence Thinking MoreQuestions
1. Answer these questions about PQRS below given that PQ = RS and PR = QS:
a
b
c
Prove ∆PRS / ∆SQP.
Prove PQRS is a parallelogram (PQ || RS and PS || QR).
Prove that the opposite angles of a parallelogram are equal.
P
R S
Q
SERIES TOPIC
J 19100% Similarity and Congruence
Mathletics 100% © 3P Learning
Similarity and Congruence
10
Questions Thinking More
2. ∆KLM is an iscosceles triangle with KL KM= .
a
b
c
KN has been drawn to bisect ML. Show that ∆KMN / ∆KLN.
Show that +MNK = +LNK = 90c .
Prove that +M = +L.
3. In ∆DEF, +F = +E, if the line GD bisects +FDE. Prove ∆DEF is isosceles.
K
NLM
EG
D
F
SERIES TOPIC
J 1020 100% Similarity and Congruence
Mathletics 100% © 3P Learning
Similarity and Congruence Thinking MoreQuestions
4. Prove the following about the Rhombus STUV below:
a
b
c
d
∆VOS / ∆TOU
∆SOT / ∆UOV
Show that the diagonals bisect each other.
Show that the diagonals bisect at 90c .
S
T
U
O
V
SERIES TOPIC
J 21100% Similarity and Congruence
Mathletics 100% © 3P Learning
Similarity and Congruence
10
Answers
Basics: Knowing More:
Using Our Knowledge:
Knowing More:
2.
1.
1.
1.
2.
3.
cmPQ 12= cmQR 18=
cmRS 20=
a
cmQR 5=
cmPT 2= cmST 4=b
P 85c+ =
S 95c+ = Q 110c+ =
D 100c+ = P 110c+ =
C 50c+ =
cmLM 10= cmAB 232=
cmKN 10=
c
d
M 45c+ = A 40c+ =
/ 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).
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.
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.
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.
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.
||| symbol 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).
a
a
b
c
d
e
f
b
Scale factor from ∆RST to ∆UVW is 51
Scale factor from ∆UVW to ∆RST is 5
Scale factor from ∆ACD to ∆ABE is 32
Scale factor from ∆ABE to ∆ACD is 121
b cmKL 6=
c cmML 12=
b
c
cmGI 15=
cmIJ 20=
d Scale factor from ∆GFH to ∆GIJ is 221
SERIES TOPIC
J 1022 100% Similarity and Congruence
Mathletics 100% © 3P Learning
Similarity and Congruence Notes
SERIES TOPIC
J 23100% Similarity and Congruence
Mathletics 100% © 3P Learning
Similarity and Congruence
10
Notes
SERIES TOPIC
J 1024 100% Similarity and Congruence
Mathletics 100% © 3P Learning
Similarity and Congruence Notes
www.mathletics.com
Similarity and Congruence
Top Related