Download 2.4 Reasoning with Properties from Algebra Algebraic Properties of

2.4 Reasoning with Properties from Algebra
Algebraic Properties of Equality
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, then a/c = b/c (c ≠ 0)
Reflexive Property For any real number a, 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 a can be substituted for b in any
equation or expression. Handout with other properties students are expected to KNOW!!!!!
Solve 5x ­ 18 = 3x + 2 and write a reason for each step.
1) 5x ­ 18 = 3x + 2 1) Given
2) 2x ­ 18 = 2 2) Subtraction Property of Equality
3) 2x = 20 3) Addition 4) x = 10
4) Division Solve 55x ­ 3(9x+ 12) = ­64 and write a reason for each step.
Solve for r and give a reason for each step.
a = 220 ­ (10/7)r
1
If (1/2x) ­ 4 = 8, then x = 24. Give a reason for each step.
1) (1/2)x ­ 4 = 8 1)
2) (1/2)x ­4 + 4 = 8 + 4 2)
3) (1/2)x + 0 = 8 + 4 3)
4) (1/2)x = 8 + 4
4)
5) (1/2)x = 12 5)
6) 2 (1/2)x = 2 ∙ 12
6)
7) 1x = 2 12
7)
8) x = 2 12 8)
9) x = 24
9) You are not expected to come up with this detailed proof, but you are expected to be able to fill in the blank for this
type of proof. 2
Properties of Length (Rules for Segments) and Measure (Rules for Angles)
Segment Length Angle Measure
Reflexive
For any segment AB, For any angle A,
AB = AB. m<A = m<A.
(This says that a segment or an angle is always equal to itself.) Symmetric
If AB = CD, then If m<A = m<B, then CD = AB.
m<B = m<A.
If the measure of two segments or angles are equal,
then you can reverse the order around the equal sign.
Transitive If AB = CD and CD = EF, If m<A = m<B and
then AB = EF. m<B = m<C, then
m<A = m<C.
ex) 2 + 6 = 3 + 5 and 3 + 5 = 1 + 7, then
2 + 6 = 1 + 7.
ex) AB = CD Give an argument that shows AC = BD. A B C D
1) AB = CD 1) Given
2) AB + BC = BC + CD 2)
3) AB + BC = AC 3)
4) BC + CD = BD
4)
5) AC = BD
5)
3
Given: m<1 + m<2 = 66
m<1 + m<2 + m<3 = 99
m<3 = m<1
m<1 = m<4
Find the m<4 and give a reason for your thinking. 1) m<1 + m<2 = 66 1) Given
m<1 + m<2 + m<3 = 99 m<3 = m<1
m<1 = m<4
2) 66 + m<3 = 99
2)
3) m<3 = 33
3)
4) m<3 = m<4
4)
5) m<4 = 33
`
5)
When you are giving reasons for a logical argument, don't overlook the
given information. This will always be your first step in a proof. I always
list all my given in the first step. The book does not do this. They some­
times put given steps in the middle of an argument. Either way is acceptable. 4