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2.5 Algebraic Proof
Monty Python’s Crazy Logic
(click on the image to view video)
2.5 Algebraic Proof
Objectives:Review properties of equality and use them to write algebraic proofs.Identify properties of equality and congruence.
Proof: An argument that uses logic, definitions, properties, and previously proven statements to show that a conclusion is true.
Section 2-5: Reasoning in AlgebraStandard: apply reflective, transitive, or symmetric properties of equality or congruence
Objectives:• Connect reasoning in algebra and geometry• Justify steps in deductive reasoning
In geometry • postulates, definitions, & properties are accepted as true (refer to page 842 for a complete list of postulates)• you use deductive reasoning to prove other statements
We will look at some basic properties used to justify statements…..
….. which leads to writing proofs.
Addition Property of Equality
If a = b, then a + c = b + c Add same amount to both sides of an equation.
Subtraction Property of Equality
If a = b, then a - c = b - c Subtract same amount to both sides of an equation.
Multiplication Property of Equality
If a = b, then a ∙ c = b ∙ c Multiply both sides of an equation by the same amount.
Division Property of Equality
If a = b and c 0, then
Divide both sides of an equation by the same amount.
a b
c c
Reflective Property of Equality
a = a Ex: 5 = 5
Symmetric Property of Equality
If a = b, then b = a Ex: 3 = 2 + 1 and 2 + 1 = 3 are the same.
Transitive Property of Equality
If a = b and b = c, then a = c.EX: If 3 + 4 = 7 and 5 + 2 = 7, then 3 + 4 = 5 + 2.
Substitution Property of Equality
If a = b , then b can replace a in any expression. Ex: a = 3; If a = b, then 3 = 3.
Distributive Property
a(b + c) = ab + ac Ex: 3(x + 3) = 3x + 9
2.5 Properties of EqualityTable on page #113
The Distributive Property states that
a(b + c) = ab + ac.
Remember!
Reflective Property of Congruence
AB AB A A
Symmetric Property of Congruence
Transitive Property of Congruence
If AB CD and AB EF, then CD EF.If A B and B C, then A C.
The Reflective, Symmetric, and Transitive Properties of Equality have corresponding properties of congruence that can be used to justify statements.
If AB CD, then CD AB.If A B, then B A
2.5 Properties of CongruenceTable on page #114
What’s the Difference between equality and congruence?
A B
AB represents the length AB, so you can think of AB as a variable representing a number.
Helpful Hint
Congruence
Geometric objects (figures / drawings) can be congruent to each other.
Equality
Measurements (numbers)can be equal to each other.
Numbers are equal (=) and
figures are congruent ().
Remember!
Statements use symbolStatements use = symbol
2.5 ApplicationWrite a justification for each step.
NO = NM + MO Segment Addition Post.
4x – 4 = 2x + (3x – 9) Substitution Property of Equality
4x – 4 = 5x – 9 Simplify.
–4 = x – 9 Subtraction Property of Equality
5 = x Addition Property of Equality
The basic format of a two column proof: Page 115
Given - facts you are given to use. STARTING POINT
Prove – conclusion you need to reach.ENDING POINT
Proof Example: Problem 3 page 116
This is given This is what you are
asked to prove
This is how you plan to get from the given to the
prove.
Reasons
Application
Statement Reason
AB + BC = AC
2y + 3y – 9 = 21
5y – 9 = 21
5y = 30
y = 6
Segment addition postulate
Substitution
Combine like terms
Addition Property (add 9 to both sides)
Division property (divide both sides by 5)
GIVEN: PROVE: y = 6
Using Properties to Justify Steps in Solving Equations
Algebra: Prove x = 43 and justify each step.
Given: m AOC = 139
m AOC = 139 Given
M AOB + m BOC = m AOC Angle Addition Postulate
x + 2x + 10 = 139 Substitution Property
3x + 10 = 139 Simplify or combine like terms
3x = 129 Subtraction Property of Equality
x = 43 Division Property of Equality
Prove : x = 43
Statement Reasons
Using Properties to Justify Steps in Solving Equations
Prove x = 20 and justify each step.
Given: LM bisects KLN
LM bisects KLN Given
MLN = KLM
4x = 2x + 40
2x = 40
x = 20
Def of Angle Bisector
Substitution Property
Subtraction Property of Equality
Division Property of Equality
Prove: x = 20
Statement Reasons
Using Properties to Justify Steps in Solving Equations
Solve for y and justify each step
Given: AC = 21
AC = 21 Given
AB + BC = AC Segment Addition Postulate
2y + 3y - 9 = 21 Substitution Property
5y – 9 = 21 Simplify
5y = 30 Addition Property of Equality
y = 6 Division Property of Equality
Find AB and BC by substituting y = 6 into the expressions.
Prove : y = 6
Now you try
Statement Reasons
Using Properties of Equality and Congruence
Name the property of congruence or equality the justifies each statement.
a. K K Reflective Property of Congruence
Symmetric Property of Equality
b. If 2x – 8 = 10, then 2x = 18Addition Property of Equality
c. If RS TW and
TW PQ,
then RS PQ.
Transitive Property of Congruence
d. If m A = m B, then
m B = m A
Use what you know about transitive properties to complete the following:
The Transitive Property of Falling Dominoes:
If domino A causes domino B to fall, and domino B causes domino C to fall, then domino A causes domino _______ to fall.
C
HOMEWORK
COMPLETE 2-5 PACKET
DUE THURSDAY NOV 1