1. 2.5 – Reasoning Using Properties of Algebra

2. If a = b, then a + c = b + c If a = b, then a – c = b – c
Name
Property of Equality
Addition Property
Subtraction Property
Multiplication Property
Division Property
Reflexive Property
Symmetric Property
Transitive Property
Substitution Property
If a = b, then a + c = b + c
If a = b, then a – c = b – c
If a = b, then ac = bc
If a = b, then a/c = b/c
For any real #, a = a
If a = b, then b = a
If a = b and b = c, then a = c
If a = b, then b can be substituted in for a

3. PROOFS!
All proofs start and end the same. It is very helpful to have a picture to refer to. We are given information and then are told to prove something.

4. Format of Proofs
Given:
Picture
Prove:
Statements Reasons 1. 2. 3. …
What you are given 1. 2. 3. …
Given
To Prove

5. Use the property to complete the statement.
2. Addition Property of Equality:
If RS = TU, then RS + 20 = ______________.
Multiplication Property of Equality:
If m1 = m2, then 3m1 = ____________.
Substitution Property of Equality:
If a = 20, then 5a = __________.
TU + 20
3m2
5(20) = 100

6. Reflexive Property of Equality:
If x is a real number, then x = _____.
Symmetric Property of Equality:
If AB = CD, then CD = __________.
Transitive Property of Equality:
If mE = mF and mF = mG, then
________________. x AB
mE = mG

7. given Addition Prop Division Prop Statements Reasons 1. 8x – 34 = 6
Complete the logical argument by writing a reason for each step.
Statements
Reasons
x – 34 = 6
1. __________________
x = 40
2. __________________
x = 5
3. __________________
given
Addition Prop
Division Prop

8. given Subtraction Prop Addition Prop Division Prop Statements Reasons
Complete the logical argument by writing a reason for each step.
Statements
Reasons
x – 7 = 6x + 7
1. __________________
x – 7 = 7
2. __________________
x = 14
3. __________________
x = -7
4. __________________
given
Subtraction Prop
Addition Prop
Division Prop

9. given Distributive Prop Subtraction Prop Addition Prop Statements
Complete the logical argument by writing a reason for each step.
Statements
Reasons
(x – 3) = 4(x + 2)
1. ___________________
x – 15 = 4x + 8
2. __________________
x – 15 = 8
3. ___________________
x = 23
4. ___________________
given
Distributive Prop
Subtraction Prop
Addition Prop

10. 1. 4x – 7 = 6x + 7 1. Given 2. – 7 = 2x + 7 2. Subtraction Prop. 3.
9. Solve the equation. Write a reason for each step
4x – 7 = 6x + 7
Statements
Reasons 1.
4x – 7 = 6x + 7 1.
Given 2.
– 7 = 2x + 7 2.
Subtraction Prop. 3.
– 14 = 2x 3.
Subtraction Prop. 4.
–7 = x 4.
Division Prop.

11. 1. 12x – 3y = 30 1. Given 2. –3y = -12x + 30 2. Subtraction Prop. 3.
10. Solve the equation for y. Write a reason for each step.
12x – 3y = 30
Statements
Reasons 1.
12x – 3y = 30 1.
Given 2.
–3y = -12x + 30 2.
Subtraction Prop. 3.
y = 4x – 10 3.
Division Prop.

12. Complete the proof by writing a reason for each step.
GIVEN: AL = SK
PROVE: AS = LK
Statements
Reasons
1. AL = SK
1. ___________________
2. LS = LS
2. ___________________
3. AL + LS = SK + LS
3. __________________
4. AL + LS = AS
4. __________________
5. SK + LS = LK
5. ___________________
6. AS = LK
6. ___________________
given
Reflexive Prop
Addition Prop
Segment Addition Prop
Segment Addition Prop
Substitution Prop

13. Steps for Proofs: 1. 2. 3. 4.
Start with given and mark pictures
What do the givens tell us?
What else do we know by looking at the picture?
End the proof with what we are trying to prove