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WarmUp
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Warm Up Answers
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2.5 Algebraic Proof
Monty Python’s Crazy Logic
(click on the image to view video)
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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.
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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.
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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
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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
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2.5 Properties of EqualityTable on page #113
The Distributive Property states that
a(b + c) = ab + ac.
Remember!
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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
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2.5 Properties of CongruenceTable on page #114
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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HOMEWORK
COMPLETE 2-5 PACKET
DUE THURSDAY NOV 1