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
• 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
• 206 = 11x + 96
• 206 - 96 = 11x + 96 - 96
• 110 = 11x + 0
• 110 = 11x
• 110/11=11x/11
• 10 = 1x
• 10 = x
• x = 10
Proof/Reasons
• 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.
Given: AB bisects RAN
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
x (2x-75) R
A
N
B
Properties of Congruence
• Transitive Property of Congruence
• Reflexive Property of Congruence
• Symmetric 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
• If a b, then b a
• If AB CD, then CD AB.
Using Properties of Equality and Congruence
Statements
• A. ST ST
• B. If m R=m S and m S=m T, then m R=m T
• C. If AB=EF, then EF=AB
Reasons/Proof
• Reflexive Property of Congruence
• Transitive Property of Equality
• Symmetric of Equality
What is the name of the property that justifies the next step?
Statements
• 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
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.
• JKL MKN
• JKM LKN
Theorem 2-4
• If 1 and 2 are right angles then 1 2.
1
2
Proving Angles are Congruent
Congruent Supplements theorem
• If 1 and 3 are supplements and 2 and
3 supplements, then 1 2.
Congruent Complements Theorem
• If 4 and 5 are complements and 5 and
6 complements then 4 6
1 2 3
4
5
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?