Download Algebra Properties and Deductive Reasoning notes.notebook

Algebra Properties and Deductive Reasoning notes.notebook
November 22, 2011
Algebra Properties (2.9)
and Deductive Reasoning (2.10)
Review of Algebra Properties Properties of Equality
Addition Property:
add the same amount to both sides
Subtraction Property :
subtract the same amount from both sides
Multiplication Property :
multiply by the same amount on both sides
Division Property :
divide both sides by the same amount
Substitution Property:
replace a value with an equal value
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Algebra Properties and Deductive Reasoning notes.notebook
November 22, 2011
Other useful properties
Reflexive :
a = a " something is equal to itself "
Symmetric : If a = b , then b = a " state an equality either way "
Transitive:
If a = b and b = c, then a = c
The properties of equality also relate to properties of congruence.
For any segments AB, CD, and EF.
For any angles R, S, and T.
Reflexive
Property
Symmetric
Property
Transitive
Property
AB
R
If AB
If R
If AB
If R
AB
R
CD , then CD
AB
S , then S R
CD, and CD
EF, then AB EF.
T.
S, and S T, then R
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Algebra Properties and Deductive Reasoning notes.notebook
Theorem
Proof November 22, 2011
statement accepted true with proof
the use of deductive reasoning
Deductive Reasoning You assume the hypothesis (given) is true and then write a series of statements with support (reasons) to lead to the conclusion (prove).
Reasons given, definitions, postulates,and previously proven theorems and algebra properties
Recall:
Postulate statement accepted without proof
Inductive reasoning using a pattern to make a conjecture
Midpoint Theorem
If point M is a midpoint of AB, then AM = AB
hypothesis (given)
Conclusion (prove)
A
statement reason
1. M is the midpoint of AB
2. AM = MB
3. AM + MB = AB
4. AM + AM = AB 2AM = AB 5. AM = AB
M
B
1. given
2. definition of midpoint
3. segment addition postulate
4. substitution property
5. division property
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Algebra Properties and Deductive Reasoning notes.notebook
November 22, 2011
Congruent Supplement Theorem
Supplements of congruent angles are congruent.
"If 2 angles are congruent, then their supplements are congruent."
Given ≅ <B
<A Statement Reason
1. <A <B
A
C
1. Given
2. m<A + m<C = 1800
m<B + m<D = 1800
2. def. of supp. angles
3. m<A + m<C = m<B + m<D
3. substitution
4. m<A + m<C = m<A + m<D
4. substitution
5. m<C = m<D
5. Subtraction property B
D
Right Angle Theorem
All right angles are congruent.
"If 2 angles are right angles then they are congruent."
Given <A and <B are right angles.
PROOF
A
B
statement reason
2. m<A = 90 , m<B = 90
1. given
2. definition of right angle
3. <A <B
3. substitution property
1. <A and <B are right angles.
0
0
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Algebra Properties and Deductive Reasoning notes.notebook
November 22, 2011
HOMEWORK 2 WKS
PROVE: Angle Bisector Theorem
A
(proof is very similar to the midpoint theorem proof from notes)
If BX is the bisector of <ABC, then m<ABX = m<ABC
X
B
C
PROVE: Congruent Complement Theorem
(proof is very similar to the congruent supplement theorem proof from notes)
Complements of congruent angles are congruent.
Given: A≅ B
D
A
C
B
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