1. Lesson 2 – 6 Algebraic Proof
Geometry
Lesson 2 – 6
Algebraic Proof
Objective:
Use algebra to write two-column proofs.
Use properties of equality to write geometric proofs.

2. Algebraic properties Addition Property of Equality
If a = b, then a + c = b + c
Subtraction Property of Equality
If a = b, then a – c = b – c
Multiplication Property of Equality
If a = b, then a(c) = b(c)
Division property
If a = b, the a/c = b/c c cannot be 0

3. Reflexive Symmetric Transitive a = a If a = b, then b = a
If a = b and b = c, then a = c

4. Distributive Property
Substitution
If a = b, then a may be replaced by b in any equation or expression.
Distributive Property
a(b + c) = ab + ac

5. Algebraic proof
A proof that is made up of a series of algebraic statements.

6. Distributive property -5x - 20 = 70 +20 +20 Subtraction prop -5x = 90
Prove that if –5(x+4) = 70, then x = -18 Write a justification for each step.
Proof:
-5(x + 4) = 70
Given
Distributive property
-5x - 20 = 70
+20
+20
Subtraction prop
-5x = 90
Substitution
Note:
Must rewrite
When asked
to show steps
Division prop.
Substitution
x = -18

7. State the property that justifies each statement.
If 4 + (-5) = -1, then x (-5) = x – 1
If 5 = y, then y = 5.
Addition property
Symmetric property

8. Solve 2(5 – 3a) – 4(a + 7) = 92. Write a Justification for each step.
Given
2(5 – 3a) – 4(a + 7) = 92
10 – 6a – 4a – 28 = 92
Distributive prop
-18 – 10a = 92
Sub.
Add. prop
+18
+18
sub
-10a = 110
Division prop
sub
a = -11

9. Two – Column Proof
Statements and reasons organized in to columns.

10. Real world problem
If the formula to convert a Fahrenheit temperature to a Celsius temperature is ,then the formula to convert a Celsius temperature to a Fahrenheit temperature is Write a two-column proof to verify this conjecture.

11. Given: Prove: Given Sub Sub Symmetric Statements Reasons 1. 2. 3. 4.5. 6.
Given
Multiplication prop
Sub
Addition prop
Sub
Symmetric

12. Write a two-column proof to verify.

13. Given: Prove: x = 3 Statements Reasons 1. Given: 1. 2. Add. Prop. 2.
3. Sub 3.
4. Mult. Prop. 4. 5.
5x + 1 = 16
5. Sub 6.
5x + 1 – 1 =
6. Subt. Prop. 7.
5x = 15
7. Sub 8.
8. Division Prop. 9.
x = 3
9. Sub

14. Properties

15. Write a two column proof to verify the conjecture.

16. Given: Prove: Statements Reasons 1. 1. Given Definition of
Congruent angles 2.
Prove: 3.
3. Transitive
x = 6 4.
6x + 7 = 8x - 5
4. Sub 5.
6x + 7 – 8x =8x – 5 – 8x
5. Subt. Prop. 6.
-2x + 7 = -5
6. Sub 7.
-2x + 7 – 7 =
7. Subt. Prop. 8.
-2x = -12
8. Sub
You do not need the
Picture This is just so
we can see it 9.
9. Division Prop.
10.
x = 6
10. Sub

17. What if you had Solved the equation differently? Have to have
Statements
Reasons 1.
1. Given
What if you had
Solved the
equation
differently?
Definition of
Congruent angles 2. 3.
3. Transitive 4.
6x + 7 = 8x - 5
4. Sub 5.
6x+7 – 6x = 8x – 5 - 6x
5. Subt. Prop. 6.
7 = 2x - 5
6. Sub 7.
7 + 5 = 2x – 5 + 5
7. Add. Prop. 8.
12 = 2x
8. Sub 9.
9. Division Prop.
Have to have
this step!
10.
6 = x
10. Sub
11. x = 6
11. Symmetric

18. Homework
Pg – 5 all, 10 – 18 E, 24