Download G Notes 2.5 Reasoning in Algebra and Geometry

2.5 Reasoning in Algebra
and Geometry
Properties of Equality Let a, b, & c be any real numbers
Addition Property :
If a  b, then a  c  b  c
Subtraction Property :
If a  b, then a  c  b  c
Multiplication Property :
If a  b, then a  c  b  c
Division Property :
a b
If a  b and c  0,then 
c c
Properties of Equality
Reflexive Property :
aa
Symmetric Property :
If a  b, then b  a
Transitive Property :
If a  b and b  c, then a  c
Substitution Property :
If a  b then b can replace a in any expression
Name the property that justifies going from the first
statement to the second.
E1
6+2=8 and 8=4+4
6+2=4+4
transitive
substitution
E2
5=1+4
1+4=5
symmetric
E3
x=3
5x = 5(3)
substitution
E4
x–3=7
x = 10
addition
E5
6x = 18
x=3
division
Distributive Property :
a b  c   a b  c   ab  ac
a b  c   a b  c   ab  ac
Proof :
a convincing argument that uses deductive reasoning
Two Column Proof :
Given : 2x + 3 = 9
Statements
1. 2x + 3 = 9
2. 2x = 6
3. x = 3
lists each statement on the left with
the justification for each statement on the right
Prove : x = 3
Reasons
1. Given
2. Subtraction Property of Equality
3. Division Property of Equality
Given : 4(x + 1) = 16
Statements
1. 4(x + 1) = 16
2. 4x + 4 = 16
3. 4x = 12
4. x = 3
Given : XY = 21
Prove : x = 3
Reasons
1. Given
2. Distributive
3. Subtraction
4. Division
2(x+3)
Prove : x = 3
X
Statements
1. XZ + ZY = XY
2. 2(x+3) + 3x = 21
3. 2x + 6 +3x = 21
4. 5x +6 = 21
5. 5x = 15
6. x = 3
Reasons
1. Segment addition property
2. Substitution
3. Distributive
4. Addition
5. Subtraction
6. Division
3x
Z
Y
PWS 2.5