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, then
a+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, then
a–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, then
ac = 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
Multiplication
Property of
Equality
Subtraction
Property of
Equality
If a=b and c0,
then
a
c

b
c
(div. prop. =)
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
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, then
b=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
21 + 7 = 4x – 7 + 7
28 = 4x
28
4

4x
4
7=x
x=7
Given equation
Add. prop. =
Simplify
Div. prop. =
Simplify
Sym. 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 Property
of Congruence
fig. A  fig. A
(refl. prop. )
Symmetric
Property of
Congruence
Transitive Property
of Congruence
Reflexive
Property of
Congruence
If fig. A  fig. B,
then fig.B 
fig.A
(sym. prop. )
Transitive Property
of 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.
5y+6
T
TA  AR
TA = AR
5y+6 = 2y+21
3y+6 = 21
3y = 15
y=5

2y+21
A
R
Given
Def.  segments
Subst. prop. =
Subtr. prop. =
Subtr. prop. =
Div. prop. =