Section 2-2:

Properties from Algebra

Properties of Equality

Addition PropertyIf a = b and c = d, then _________________________.

Subtraction Property

If a = b and c = d, then __________________________. Multiplication Property

If a = b, then _________________________________.

Division PropertyIf a = b and c ≠ 0, then ________________________.

a + c = b + d

a – c = b – d

ca = cb

𝑎𝑐

=𝑏𝑐

Ex: If x = 12, then x + 2 = 14. If x – 3 = 7, then x = 10.

Ex: If x + 2 = 9, then x = 7.

Ex: If = 9, then x = 27.

Ex: If 4x = 28, then x = 7.

Substitution Property

If a = b, then either a or b may be ____________ for the other in any equation (or inequality).

substituted

Ex: If mA = 30° and mA = mC, then mC = 30°. Ex: If 2x + 3 = y and x = 5, then 13 = y.

Ex: If x = y and z = y, then x = z.

Reflexive Propertya = _____

Symmetric Property

If a = b, then ____________________. Transitive Property

If a = b and b = c, then _____________.

Ex: If mA, then mA. a

b = a

a = c

Ex: If AB = CD, then CD = AB.

Ex: If mA = mB and mB = mC, then mA = mC.

Properties of Congruence

Reflexive Property ________ D _________

Symmetric PropertyIf , then _____________________________.

If D E, then ______________________________.

Transitive Property

If and , then _______________.

If D E and E F, then _________________.

𝐷𝐸 D

D

Properties of Real Numbers

Commutative Propertya + b = __________, ab = _______ Associative Property a + (b + c) = _________, a(bc) = ________ Distributive Propertya(b + c) = __________

b + a ba

(a + b) + c (ab)c

ab + ac

Examples:Justify each step with a “Property from Algebra.”Follow the example below: Given: 4x – 5 = –2Prove: x =

Statements Reasons

1. 4x – 5 = –2 1. Given 2. 4x = 3 2. Addition Property of Equality 3. x = 3. Division Property of Equality

1. Given: Prove: a =

Statements Reasons 1. 1. Given 2. 3a = 2. 3. a = 3.

Multiplication Property of Equality

Division Property of Equality

2. Given: –11 Prove: z = –40

Statements Reasons

1. –11 1. Given 2. z + 7 = –33 2. 3. z = –40 3.

Multiplication Property of Equality

Subtraction Property of Equality

3. Given: 15y + 7 = 12 – 20y Prove: y =

Statements Reasons

1. 15y + 7 = 12 – 20y 1. Given 2. 35y + 7 = 12 2. 3. 35y = 5 3.

4. y = 4.

Addition Property of EqualitySubtraction Property of EqualityDivision Property of Equality

4. Given: x – 2 = Prove: x = 6

Statements Reasons

1. x – 2 = 1. Given 2. 5(x – 2) = 2x + 8 2. 3. 5x – 10 = 2x + 8 3. 4. 3x – 10 = 8 4. 5. 3x = 18 5. 6. x = 6 6.

Multiplication Property of Equality

Distributive PropertySubtraction Property of EqualityAddition Property of EqualityDivision Property of Equality

Substitution Property Practice

I. 1. a = b + c 1. Given d = e + f2. a = d 2. Given3. ____________ 3. Substitutionb + c = e + f

II.1. a = b + c 1. Given d = e + f2. b + c = e + f 2. Given3. _____________ 3. Substitutiona = d

(Diagram is for III. And IV.)III. 1. DF = AC 1. Given2. DE + EF = ____ 2. ____________

AB + BC = ____ ____________3. ____________ 3. Substitution

•

•

• •

••D E

A B C

F

DFAC

DE + EF = AB + BC

SegmentAddition Post.

IV.

1. DE + EF = AB + BC 1. Given2. DE + EF = _____ 2. ______________ AB + BC = _____ ______________3. _____________ 3. Substitution

DFAC

SegmentAddition Post.

DF = AC

V.1. WOY XOZ 1. Given2. mWOY = m1 + m2 2. _____________ mXOZ = m3 + m2 _____________

3. ________________ 3. Substitution ________________

O

WX

Y

Z•

•

•

•

•1 2

3

AngleAddition Post.

m1 + m2 =m3 + m2

CLASSWORK:page 41 #1-8 all