Download 2.5 Reason Using Properties from Algebra

2.5 Reason Using Properties
from Algebra
Objective: To use algebraic
properties in logical arguments.
Algebraic Properties
• 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 ac = bc.
• Division Property:
If a = b and c = 0, then a/c = b/c.
Algebraic Properties
• Substitution Property: If a = b, then a can be
substituted for b in an equation or
expression.
• Distributive Property: a(b + c) = ab + ac,
where a, b, and c are real numbers.
Example 1:
Write a two-column proof to solve the equation.
3x + 2 = 8
Statements
Reasons
1.
2.
3.
4.
5.
Given
3x + 2 = 8
3x + 2 – 2 = 8 – 2
3x = 6
3x ÷ 3 = 6 ÷ 3
x=2
Subtraction Prop
Simplify
Division Prop
Simplify
Example 2:
Write a two-column proof to solve the equation.
Statements
1.
2.
3.
4.
5.
6.
4x + 9 = 16 – 3x
4x + 9 + 3x = 16 – 3x + 3x
7x + 9 = 16
7x + 9 – 9 = 16 – 9
7x = 7
7x ÷ 7 = 7 ÷ 7
7. x = 1
Reasons
Given
Addition Prop
Simplify
Subtraction Prop
Simplify
Division Prop
Simplify
Example 3:
Write a two-column proof to solve the equation.
2(-x – 5) = 12
Statements
1.
2.
3.
4.
5.
2(-x – 5) = 12
-2x – 10 = 12
-2x – 10 + 10 = 12 + 10
-2x = 22
-2x ÷ -2 = 22 ÷ -2
6. x = -11
Reasons
Given
Distributive Prop
Addition Prop
Simplify
Division Prop
Simplify
Algebraic Properties
• Reflexive Property:
For any real number a, a = a
For any segment AB, AB = AB
For any angle A, m<A = m<A
• Symmetric Property:
For any real numbers a and b, if a = b, then b = a
For any segments AB and CD, if AB = CD, then CD = AB
For any angles A and B, if m<A = m<B, then m<B = m<A
Algebraic Properties (cont)
Transitive Property:
For any real numbers a, b and c, if
a = b and b = c, then a = c.
For any segments AB, CD, and EF, if
AB = CD and CD = EF, then AB = EF.
For any angles A, B and C, if
m<A = m<B, and m<B = m<C then m<A = m<C
Example 4
In the diagram, AB = CD. Show that AC = BD.
Statement
AB = CD
Reason
Given
AC = AB + BC
Segment Addition Postulate
BD = BC + CD
Segment Addition Postulate
AB + BC = CD + BC
AC = BD
Addition Property of Equality
Substitution Property of Equality
Example 5
You are designing a logo to
sell daffodils. Use the
information given. Determine
whether m EBA = m DBC.
Statement
m
Reason
1=m
Given
3
m
EBA = m
3+ m
2
m
EBA = m
1+ m
2
m
1+m
m
2 =m
EBA = m
DBC
DBC
Angle Addition Postulate
Substitution Property of Equality
Angle Addition Postulate
Transitive Property of Equality
Example5:6
Example
Name the property of equality the statement illustrates.
a). If m
6=m
7, then m
7=m
6.
ANSWER
Symmetric Property of Equality
b). If JK = KL and KL = 12, then JK = 12.
ANSWER
Transitive Property of Equality
Example 5 cont’d:
c). m
W=m
W
ANSWER
Reflexive Property of Equality
d). If L = M and M = 6, then L = 6
ANSWER
Transitive Property of Equality