Download 2.5 Reasoning in Algebra and Geometry

2.5 Reasoning in
Algebra and
Geometry and 2.6
Proving Angles are
Congruent
Lesson Purpose
Objective
• To connect reasoning in
algebra and geometry
Essential Question
• How can you make a
conjecture and prove it is
true?
ALGEBRAIC PROPERTIES OF EQUALITY
• Addition Property of Equality
• Subtraction Property of Equality
• Multiplication Property of Equality
• Division Property of Equality
• Substitution Property of Equality
• Transitive Property of Equality
• Reflexive Property of Equality
• Symmetric Property of Equality
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If a = b, then a + c = b + c.
If a = b, then a – c = b – c.
If a = b, then a c = b c.
If a = b, then a/c = b/c.
If a = b, then you may replace
b with a in any expression.
• If a = b and b = c, then a = c.
• a=a
• If a = b, then b = a
Vocabulary Definitions
• Proof :
– Convincing argument
that uses deductive
reasoning
• Theorem:
– Is a conjecture or
statement that you
prove is true.
• Two Column Proof:
– Lists each statements on
left and the proof on the
right
Example: Justify Steps when Solving an
Equation
Statements
Proof/Reasons
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206 = 11x + 96
206 - 96 = 11x + 96 - 96
110 = 11x + 0
110 = 11x
110/11=11x/11
10 = 1x
10 = x
x = 10
Given
Subtraction Property of Equality
Subtraction
Additive Identity Property
Division Property of Equality
Division
Multiplicative Identity Property
Symmetric Property
What is the value of x? Justify each step.
B
R
Given: AB bisects
RAN
(2x-75)
x
N
A
Statements:
• X= 2x-75
• x+75=2x
• 75=2x-x
• 75=x
Proof:
• Definition of an bisector
• Addition Property of Equality
• Subtraction Prop. Of equality
• Distribution Property
Properties of Congruence
• Transitive Property of
Congruence
• Reflexive Property of
Congruence
• If a
b and b
c, then
a
c.
• If AB CD and CD EF, then
AB EF.
•
a
a; AB BA
• Symmetric Property of
Congruence
• If a
b, then b
a
• If AB CD, then CD AB.
Using Properties of Equality and Congruence
Statements
Reasons/Proof
• A. ST ST
• Reflexive Property of
Congruence
• B. If m R=m S and
• Transitive Property of
m S=m T, then m R=m T
Equality
• C. If AB=EF, then EF=AB
• Symmetric of Equality
What is the name of the property that justifies the
next step?
Statements
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A. AR TY
TY AR
B. 3(x+5)=9
3x+15=9
C. ¼ x=7
x=28
D. m R =m R
Proof/Reasons
• Sym Property of
• Distributive Property
• Multiplication Property of
Equality
• Reflective Property of
Equality
Write a Two Column Proof
Given: AB CD
Prove AC BD
Statements:
• AB CD
• AB+BC=BC+CD
• AB+BC=AC or BC+CD=BD
• AC=BD
• AC BD
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A
B
C
D
Proof:
Given
Addition Property of =
Segment Addition Property
Substitution Property of =
Segments that = are
Proving Angles are Congruent
Vertical Angle Theorem
• Vertical Angles are
congruent.
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JKL
JKM
MKN
LKN
Theorem 2-4
• If 1 and 2 are right
angles then 1 2.
2
1
Proving Angles are Congruent
Congruent Supplements
theorem
Congruent Complements
Theorem
• If 1 and 3 are
supplements and 2 and
3 supplements, then
1 2.
• If 4 and 5 are
complements and 5 and
6 complements then
4 6
4
5
1
2
3
6
Recap: Summary
• You use deductive reasoning and properties to solve
equations and justify your reasoning.
• A proof is a convincing argument that uses deductive
reasoning. A two column proof lists each statement
on the left and the justification for each statement
on the right.
Ticket Out
• What is the main difference
between a property of
equality and a property of
congruence?