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2.6
algebraic
proofs
Page 75
Lesson Objectives
Standards
Lesson Notes
2.6 Algebraic Proofs
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Lesson Objectives
Standards
Lesson Notes
After this lesson, you should be able to successfully apply algebraic properties to write proofs. Press the tabs to view details.
A list of algebraic steps to solve problems where each step is justified is called an algebraic proof, The table shows properties you have studied in algebra.
September 14, 2016
Lesson Objectives
Standards
Lesson Notes
G.CO.9 Prove theorems about lines and angles.
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Addition Property of
Equality
Subtraction Property
of Equality
If a = b, then
a + c = b + c
If a = b, then
a - c = b - c
Multiplication Property Division Property of
of Equality
Equality
If a = b, then
ac = bc
If a = b and c = 0, then
a
b
=
c
c
Reflexive Property
of Equality
Symmetric Property
of Equality
a = a
If a = b, then b = a
Transitive Property
of Equality
If a = b and b = c,
then a = c
Substitution Property
of Equality
If a = b, then a may
be replaced by b in
any equation
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Example 1: Solve 2x + 3 = 9 − x. Write a reason for each step.
Distributive Property:
a(b + c) = ab + bc
Equation
Explanation
2x + 3 = 9 ­ x Rewrite the original equation
2x + 3 + ____ = 9 ­ x + ____
Add _____ to each side
Reason
Given
Addition
Property
Substitution
________ + 3 = ________ Combine like terms __________ = __________
Subtract _____ from each side
x = _____
Divide each side by _____
Subtraction Property
Division Property
Example 2: Solve ­4(6x + 2) = 64. Write a reason for each step.
Equation
Explanation
­4(6x + 2) = 64 Rewrite the original equation
__________________ = 64
Multiply
_________ = _________
Add _____ to each side
x = _____
Divide each side by _____
Reason
Given
Addition
Property
Division Property
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In geometry, a similar format is usedà two column proof or formal proof
Statements and reasons organized in two columns
Each step is called ________________
Statement
Properties that justify each step are called _____________
Reason
Geometric Proof Geometry deals with numbers as measures, so geometric proofs use properties of numbers. Here are some of the algebraic properties used in geometric proofs.
Property
Segments
Angles
Reflexive
AB = AB
m<1 = m<1
Symmetric
If AB = CD, then CD = AB.
If m<1 = m<2, then
m<2 = m<1.
Transitive
If AB = CD and CD = EF, If m<1 = m<2 and m<2 = then AB = EF.
m<3, then m<1 = m<3.
Example 3: Write a two­column proof to verify this conjecture.
Given: mÚ1 = mÚ2, mÚ2 = mÚ3
Prove: mÚ1 = mÚ3
Statements
Reasons
1. mÚ1 = mÚ2
1. _____________
2. mÚ2 = mÚ3
2. _____________
3. mÚ1 = mÚ3
Transitive
3. _____________
Given
Given
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State the property that justifies each statement.
1. If m∠1 = m∠2, then m∠2 = m∠1.
Transitive Property
Symmetric Property
Reflexive Property
2. If m∠1 = 90 and m∠2 = m∠1, then m∠2 = 90.
Substitution Property
Symmetric Property
Transitive Property
3. If AB = RS and RS = WY, then AB = WY.
Transitive Property
Symmetric Property
Reflexive Property
4. If AB = CD, then Substitution Property
Addition Property Multiplication Property
5. If m∠1 + m∠2 = 110 and m∠2 = m∠3, then m∠1 + m∠3 = 110.
Symmetric Property Substitution Property Transitive Property
6. RS = RS
Transitive Property
Symmetric Property Reflexive Property
7. If AB = RS and TU = WY, then AB + TU = RS + WY.
Substitution Property
Addition Property Multiplication Property
8. If m∠1 = m∠2 and m∠2 = m∠3, then m∠1 = m∠3.
Transitive Property
Symmetric Property
Substitution Property
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Example 3: Solve 6x + 2(x − 1) = 30. Write a justification for each step.
Given
Distributive Property
Substitution
Addition Property
Substitution
Division Property
Substitution
Extra problem
More Practice – 2.6 State the property that justifies each statement.
1. If 80 = mÚA, then mÚA = 80.
Practice
2. If RS = TU and TU = YP, then RS = YP.
3. If 7x = 28, then x = 4.
4. If VR + TY = EN + TY, then VR = EN.
5. If mÚ1 = 30 and mÚ1 = mÚ2, then mÚ2 = 30. 6
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6.
September 14, 2016
Complete the following proof.
Given: 8x − 5 = 2x + 1 Prove: x = 1
9. Given: 4x + 8 = x + 2
Prove: x = −2
Prove: x = 3
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Answers:
1. Symmetric 3. Division 5. Substitution 7. J, D, E, G, H, A, C, F, I, B 9. Given, subtraction, 3x + 8 – 8 = 2 – 8, 3x = −6, division, x = −2 + BOOK WORK p. 137(9­18, 43, 56 ­ 58)
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2.6 algebraic proofs ink.notebook
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2.6 algebraic proofs ink.notebook
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Check book work answers to the odd
problems in the back of the book
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