Algebraic Proofs Equations

Algebraic Proofs Equations - Transcript

2 6 Algebraic Proof 10 17 08
Algebra is a system with sets of numbers operations and properties that allow you to perform algebraic operations

Properties of Equality for Real Numbers
Reflexive Property a a Symmetric Property if a b then b a
Transitive Property if a b and b c then a c Addition and Subtraction Property if a b then a c b c and a c b c Multiplication and Division Property if a b then a c b c and if c 0 then a c b c Substitution Property For all numbers a and b if a b then a may be replaced by b in any equation or expression Distributive Property a b c ab ac

The properties of equality can be used to justify each step when solving an equation A group of algebraic steps used to solve problems form a deductive argument

Verify Algebraic Relationships Solve 3 x 2 42
Algebraic Steps 3 x 2 42 3x 6 42 3x 6 6 42 6 3x 48 Properties Original Equation Distributive Property Addition Property Substitution Division Substitution

3 x 48 3 3
x 16

Two Column Proofs
A two column proof or formal proof contains statements and reasons organized in two columns In a two column proof each step is called a statement The properties that justify each step are called reasons

Write a Two Column proof
Statements 1 3 5 1 x 3 5 2 3 x 3 1 3 3 4 5 6
5 If 3 thenx x 1 2 3

Reasons 1 Given

2 Distributive Property 3 4 5 6 Substitution Addition Substitution Division

3x 5 1 3x 5 5 1 5 3x 6 x6 33

7 x 2

7 Substitution

Geometric Proof
Property Reflexive Segments AB AB Angles

m m 1 1
Ifm m 1 2

Symmetric If AB CD then CD

AB

thenm m 2 1
andm m 2 3 thenm m 1 3
Ifm m 1 2

Transitive

If AB CD and CD EF then AB EF

Geometric Proof
On a clock the angle formed by the hands at 2 00 is a 60 angle If the angle formed at 2 00 is congruent to the angle formed at 10 00 prove that the angle at 10 00 is a 60 angle

2 2 10 Given m 60 o Prove m 60 10
Statements

Geometric Proof Cont

Reasons 1 Given 2 Definition of congruent angles 3 Substitution 4 Symmetric Property

2 1 m 60 2 10
2 3 4

2 10 60 m 10

m 0 60 1

Extra Practice
Classwork Study Guide 2 6 Homework Practice 2 6