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Lesson 4-2: Congruent Triangles 2
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 3
Congruent Triangles - CPCTC
If ABC PQR
CPCTC: Corresponding Parts of Congruent Triangles are Congruent
Two triangles are congruent IFF their corresponding parts (angles and sides) are congruent.
BC
A
QR
PA ↔ P; B ↔ Q; C ↔ R
Vertices of the 2 triangles correspond in the same order as the triangles are named.
Corresponding sides and angles of the two congruent triangles:
A
A P B Q C
B PQ BC Q C P
R
R A R
≡
≡
=
=
│
│
Lesson 4-2: Congruent Triangles 4
Congruent Triangles
____
____
____
AB
BC
AC
ZY
YX
B
A
C
X Y
Z
≡ ≡=
=
│ │
∆ABC ______
A ____ Z
B _____
C ______
Y
X
∆ ZYX
∆ABC ∆XYZ
Note:
ZX
ZY
Lesson 4-2: Congruent Triangles 5
When referring to congruent triangles (or polygons), we must name corresponding vertices in the same order.
R
AY
S
UN
S
U
N
R
A
YSUN RAY
Also NUS YAR
Also USN ARY
Example…………
Lesson 4-2: Congruent Triangles 6
Example ………
M
O
N
TA SR
UP
E
1. Pentagon MONTA Pentagon PERSU
2. Pentagon ATNOM Pentagon USREP
3. Etc.
If these polygons are congruent, how do you name them ?
Lesson 4-1: Using Properties 8
Commutative & Associative Property
...order does not matter.
Addition: a + b = b + a
Multiplication: a • b = b • a
4 + 5 = 5 + 4
2 • 3 = 3 • 2
Examples
The commutative and associative property does not work for subtraction or division.
Commutative Property
Associative Property ...grouping does not matter
Addition: (a + b) + c = a + (b + c)
Multiplication: (ab) c = a (bc)
(1 + 2) + 3 = 1 + (2 + 3)
(2•3)•4 = 2•(3•4)
Lesson 4-1: Using Properties 9
Properties for Addition & Multiplication
1a
a + = a
“0”is the identity element for addition 0
Additive Inverse:
a + = 0
a and (-a) are called opposites
(-a)
Multiplicative Identity “1”is the identity element for multiplication
a • = a 1
Multiplicative Inverse a and are called reciprocals
a • = 1
Additive Identity:
1a
Lesson 4-1: Using Properties 10
Multiplicative & Distributive Property
Multiplicative Property of Zero a • 0 = ___ 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
(b + c) a = b a + ca (b - c) a = ba - caand
Lesson 4-1: Using Properties 11
1) 5a + (6 + 2a) = 5a + (2a + 6)
2) 5a + (2a + 6) = (5a + 2a) + 6
3) 2(3 + a) = 6 + 2a
Commutative (switch order)
Associative (switch groups)
Distributive
Name the property :
Examples………….
Lesson 4-1: Using Properties 12
Properties of Equality
Addition
Subtraction
Multiplication
Division
If a = b, then
a + c = b + c
a - c = b - c
a • c = b • c
a / c = b / c
x 0
Substitution: If a = b, then a can be replaced by b
Example: (5 + 2)x = 7x
Lesson 4-1: Using Properties 13
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
Transitive: If ab and bc, then ac
If XY and YZ, then XZ
If C D, then D C