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Lesson 4-2: Congruent Triangles 1 Lesson 4-2 Congruent Triangles

Lesson 4-2: Congruent Triangles 1 Lesson 4-2 Congruent Triangles

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Lesson 4-2: Congruent Triangles 1

Lesson 4-2

Congruent Triangles

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 7

Lesson 4-1

Using Properties

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