GeometryLesson 2 – 6

Algebraic Proof

Objective:Use algebra to write two-column proofs.

Use properties of equality to write geometric proofs.

Algebraic propertiesAddition 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

Reflexivea = a

SymmetricIf a = b, then b = a

TransitiveIf a = b and b = c, then a = c

SubstitutionIf a = b, then a may be replaced

by b in any equation or expression.

Distributive Propertya(b + c) = ab + ac

Algebraic proof

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

Prove that if –5(x+4) = 70, then x = -18Write a justification for each step.

Proof:

-5(x + 4) = 70 Given

-5x - 20 = 70 Distributive property

+20 +20 Subtraction prop

-5x = 90 Substitution

5

90

5

5

x

Division prop.

x = -18 Substitution

Note:Must rewriteWhen asked to show steps

State the property that justifies each statement.

If 4 + (-5) = -1, then x + 4 + (-5) = x – 1

If 5 = y, then y = 5.

Addition property

Symmetric property

Solve 2(5 – 3a) – 4(a + 7) = 92. Write a Justification for each step.

2(5 – 3a) – 4(a + 7) = 92 Given

10 – 6a – 4a – 28 = 92 Distributive prop

-18 – 10a = 92 Sub.

+18 +18 Add. prop

-10a = 110 sub

a = -11

10

110

10

10

a

Division prop

sub

Two – Column Proof

Statements and reasons organized in to columns.

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.

329

5 FC

325

9 CF

Given: 329

5 FC

Prove: 325

9 CF

Statements Reasons

1. 1.

2. 2.

3. 3.

4. 4.

5. 5.

6. 6.

329

5 FC Given

329

5 FC

5

9

5

9 Multiplication prop

325

9FC Sub

3232325

9 FC Addition prop

FC 325

9 Sub

325

9 CF Symmetric

Write a two-column proof to verify.

3,082

15

then

xIf

Statements ReasonsGiven: Prove:

082

15

x

x = 3 1. 082

15

x 1. Given:

2. 80882

15

x 2. Add. Prop.

3. 82

15

x 3. Sub

4. 282

152

x 4. Mult. Prop.

5. 5x + 1 = 16 5. Sub

6. 5x + 1 – 1 = 16 - 1 6. Subt. Prop.

7. 5x = 15 7. Sub

8. 5

15

5

5x

8. Division Prop.

9. 9. Subx = 3

Properties

6

,&

xthen

KGHJGKJGKFGJIfWrite a two column proof to verify the conjecture.

Statements ReasonsGiven:

Prove:

JGKFGJ KGHJGK

76 xFGJm58 xKGHm

x = 6

1. JGKFGJ KGHJGK

1. Given

2. JGKmFGJm KGHmJGKm

2. Definition of Congruent angles

3.

You do not need the Picture This is just so we can see it

KGHmFGJm 3. Transitive

4. 4. Sub6x + 7 = 8x - 5

5. 6x + 7 – 8x =8x – 5 – 8x 5. Subt. Prop.

6. -2x + 7 = -5 6. Sub

7. -2x + 7 – 7 = -5 - 7 7. Subt. Prop.

8. -2x = -12 8. Sub

9.2

12

2

2

x

9. Division Prop.

10. x = 6 10. Sub

Statements Reasons

What if you had Solved the equationdifferently?

1. JGKFGJ KGHJGK

1. Given

2. JGKmFGJm KGHmJGKm

2. Definition of Congruent angles

3. KGHmFGJm 3. Transitive

4. 4. Sub6x + 7 = 8x - 5

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. Sub9.

2

2

2

12 x 9. Division Prop.

10. 6 = x 10. Sub

11. x = 6 11. Symmetric

Have to have this step!

Homework

Pg. 137 1 – 5 all, 10 – 18 E, 24