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Lesson 4-2
Congruent
Triangles
Lesson 4-2: Congruent Triangles
1
Congruent Figures
Congruent figures are two figures that have the same
size and shape.
IF two figures are congruent THEN they have the
same size and shape.
IF two figures have the same size and shape THEN
they are congruent.
Two figures have the same size and shape IFF they are
congruent.
Lesson 4-2: Congruent Triangles
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Congruent Triangles - CPCTC
CPCTC: Corresponding Parts of Congruent Triangles are Congruent
Two triangles are congruent IFF their corresponding parts
(angles and sides) are congruent.
A
If
ABC  PQR
A ↔ P; B ↔ Q; C ↔ R
B
C
≡
Vertices of the 2 triangles correspond in the same order
as the triangles are named.
P
Corresponding sides and angles of the two congruent triangles:
AB  PQ
B  Q
BC  QR
C  R
AC  PR
Lesson 4-2: Congruent Triangles
Q
≡
A  P
3
R
Congruent Triangles
B
C


Z
____

Y
_____

X
______
B
=
A 
ZY
A
X
=
ZX
AC  ____
Z
≡ ≡
AB  ____
ZY
YX
BC  ____
C
Note:
∆ABC

∆
ZYX
______
∆ABC

∆XYZ
Lesson 4-2: Congruent Triangles
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Y
Example…………
When referring to congruent triangles (or polygons), we must
name corresponding vertices in the same order.
R
Y
S
R
U
A
N
Y
SUN  RAY
A
N
U
Also NUS  YAR
Also USN  ARY
S
Lesson 4-2: Congruent Triangles
5
Example ………
If these polygons are congruent, how do you name them ?
P
O
U
N
M
E
A
T
S
R
1. Pentagon MONTA  Pentagon PERSU
2. Pentagon ATNOM  Pentagon USREP
3. Etc.
Lesson 4-2: Congruent Triangles
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Lesson 4-1
Using
Properties
Lesson 4-1: Using Properties
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Commutative & Associative Property
Commutative Property ...order does not matter.
Addition:
Examples
4+5=5+4
a+b=b+a
2•3=3•2
Multiplication: a • b = b • a
Associative Property ...grouping does not matter
Addition: (a + b) + c = a + (b + c)
(1 + 2) + 3 = 1 + (2 + 3)
Multiplication: (ab) c = a (bc)
(2•3)•4 = 2•(3•4)
The commutative and associative property does not work for
subtraction or division.
Lesson 4-1: Using Properties
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Properties for Addition & Multiplication
Additive Identity: “0”is the identity element for addition
a+0 =a
Additive Inverse:
a and (-a) are called opposites
a + (-a) = 0
Multiplicative Identity “1”is the identity element for multiplication
a• 1 =a
Multiplicative Inverse a and
1
a
are called reciprocals
1
a•
=1
a
Lesson 4-1: Using Properties
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Multiplicative & Distributive Property
Multiplicative Property of Zero
0
a • 0 = ___
Multiplicative Property of -1
a • -1 = ___
-a
The Distributive Property
The process of distributing the number on the outside of the
parentheses to each term on the inside.
a(b + c) = a b + ac
a(b - c) = ab - ac
and
(b + c) a = b a + ca
(b - c) a = ba - ca
Lesson 4-1: Using Properties
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Examples………….
Name the property :
1) 5a + (6 + 2a) = 5a + (2a + 6) Commutative (switch order)
2) 5a + (2a + 6) = (5a + 2a) + 6 Associative (switch groups)
3) 2(3 + a) = 6 + 2a
Distributive
Lesson 4-1: Using Properties
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Properties of Equality
If a = b, then
Addition
a+c=b+c
Subtraction
a-c=b-c
Multiplication
a•c=b•c
Division
a/c=b/c
Substitution:
If a = b, then a can be replaced by b
Example: (5 + 2)x = 7x
x0
Lesson 4-1: Using Properties
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Properties of Equality & Congruence
Reflexive: a = a
5=5
Symmetric: If a = b then b = a
If 4 = 2 + 2 then 2 + 2 = 4
Transitive: If a=b and b=c, then a=c
If 4 = 2 + 2 and 2 + 2 = 3 + 1, then 4 = 3 + 1
Reflexive: a  a
A  B
Symmetric: If a  b then b  a
If C  D, then D  C
Transitive: If ab and bc, then ac
If XY and YZ, then XZ
Lesson 4-1: Using Properties
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