Obj. 7 Algebraic Proof

proof – an argument which uses logic, definitions, properties, and previously proven statements

algebraic proof – A proof which uses algebraic properties

• When you write a proof, you must give a justification (reason) for each step to show that it is valid. For each justification, you can use a definition, postulate, property, or a piece of given information.

Algebraic Properties Foldable• Make a hotdog fold.• Make shutters.• Open up all the folds and make a hamburger

fold.• Make shutters.• Cut from the edge of the paper to the fold on

each side. This should give you eight sections.

Addition Property of

Equality

Subtraction Property of

Equality

Multiplication Property of

Equality

Division Property of

Equality

Reflexive Property of

Equality

Symmetric Property of

Equality

Transitive Property of

Equality

Substitution Property of

Equality

If a = b, thena + c = b + c(add. prop.

=)

Subtraction Property of

Equality

Multiplication Property of

Equality

Division Property of

Equality

Reflexive Property of

Equality

Symmetric Property of

Equality

Transitive Property of

Equality

Substitution Property of

Equality

Addition Property of

Equality

If a = b, thena – c = b – c(subtr. prop.

=)

Multiplication Property of

Equality

Division Property of

Equality

Reflexive Property of

Equality

Symmetric Property of

Equality

Transitive Property of

Equality

Substitution Property of

Equality

Addition Property of

Equality

Subtraction Property of

Equality

If a = b, thenac = bc

(mult. prop. =)

Division Property of

Equality

Reflexive Property of

Equality

Symmetric Property of

Equality

Transitive Property of

Equality

Substitution Property of

Equality

Addition Property of

Equality

Subtraction Property of

Equality

Multiplication Property of

Equality

If a=b and c0, then

(div. prop. =)

Reflexive Property of

Equality

Symmetric Property of

Equality

Transitive Property of

Equality

Substitution Property of

Equality

a bc c

Addition Property of

Equality

Subtraction Property of

Equality

Multiplication Property of

Equality

Division Property of

Equality

a = a(refl. prop.

=)

Symmetric Property of

Equality

Transitive Property of

Equality

Substitution Property of

Equality

Addition Property of

Equality

Subtraction Property of

Equality

Multiplication Property of

Equality

Division Property of

Equality

Reflexive Property of

Equality

If a = b, thenb = a

(sym. prop. =)

Transitive Property of

Equality

Substitution Property of

Equality

Addition Property of

Equality

Subtraction Property of

Equality

Multiplication Property of

Equality

Division Property of

Equality

Reflexive Property of

Equality

Symmetric Property of

Equality

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

c(trans. prop.

=)

Substitution Property of

Equality

Addition Property of

Equality

Subtraction Property of

Equality

Multiplication Property of

Equality

Division Property of

Equality

Reflexive Property of

Equality

Symmetric Property of

Equality

Transitive Property of

Equality

If a = b, then b can be

substituted for a(subst. prop. =)

Example: Solve the equation 21 = 4x – 7. Write a justification for each step.

21 = 4x – 7 Given equation21 + 7 = 4x – 7 + 7 Add. prop. = 28 = 4x Simplify

7 = x Simplify x = 7 Sym. prop. =

284

4x4 Div. prop. =

Line segments with equal lengths are congruent, and angles with equal measures are also congruent. Therefore, the reflexive, symmetric, and transitive properties of equality have corresponding properties of congruence.

•Hotdog fold•Open it up and hamburger fold•Make shutters•Cut one side of shutters into two sections.

Reflexive Property of Congruence

Symmetric Property of Congruence

Transitive Propertyof Congruence

fig. A fig. A(refl. prop. )

Symmetric Property of Congruence

Transitive Propertyof Congruence

Reflexive Property of Congruence

If fig. A fig. B, then fig.B

fig.A(sym. prop.

)

Transitive Propertyof Congruence

Reflexive Property of Congruence

Symmetric Property of Congruence

If fig. A fig. B and fig. B fig. C, then figure A figure

C(trans. prop. )

Example: Write a justification for each step.

Given TA = AR Def. segments5y+6 = 2y+21 Subst. prop. =3y+6 = 21 Subtr. prop. = 3y = 15 Subtr. prop. = y = 5 Div. prop. =

RAT

5y+6 2y+21

TA AR