Download 2-5: Reasoning in Algebra and Geometry Properties of Equality

2-5: Reasoning in Algebra and Geometry
Properties of Equality
Properties of Equality
Assume a, b and c represent real numbers.
Property
Definition
Example
Addition
If a  b, then a  c  b  c
If x  12, then x  3  12  3
Subtraction
If a  b, then a  c  b  c
If x  12, then x  3  12  3
Multiplication
If a  b, then a  c  b  c
If x  12, then x  3  12  3
Division
 with c  0 
If a  b, then a  c  b  c
If x  12, then x  3  12  3
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Properties of Equality
Properties of Equality
Assume a, b and c represent real numbers.
Property
Definition
Example
aa
Reflexive
55
1
1
 .5, then .5 
2
2
Symmetric
If a  b, then b  a
Transitive
If a  b and b  c, then a  c
Substitution
If a  b, then you can replace If a  b and 9  a  15, then
9  b  15
a with b and vice versa
If
If 2.5  2 1 and 2 1  5 ,
2
2
2
then 2.5  5
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Properties of Equality
Properties of Equality
Assume a, b and c represent real numbers.
Property
Distributive
Definition
Example
a  b  c   ab  ac
5  x  6   5 x  30
a  b  c   ab  ac
5  x  6   5 x  30
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Properties of Congruence
Properties of Congruence
Property
Example
Example
AB  AB
A  A
Symmetric
If AB  CD, then CD  AB
If A  B, then B  A
Transitive
If AB  CD and CD  EF ,
then AB  EF
If A  B and B  C ,
then A  C
Reflexive
2-6: Proving Angles Congruent
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Theorem

Vertical Angles Theorem
◦ Vertical Angles are congruent.
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1  3 and 2  4

3
1
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Congruent Supplements Theorem
◦ If two angles are _____________
of the same
supplements
angle (or of congruent angles), then the two
angles are ___________.
congruent
If 1 and 3 are supplements and
3
1
2 and 3 are supplements
2
then 1  2
Theorem

Congruent Complements Theorem
◦ If two angles are _____________
complements of the same
angle (or of congruent angles), then the two
angles are __________.
congruent
If 1 and 2 are complements and
1
2

3
2 and 3 are complements
then 1  3
Theorem 2-4
◦ All right angles are congruent

Theorem 2-5
◦ If two angles are congruent and supplementary,
then each is a right angle.
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Homework: p. 117 #8-12, 14-19
p. 124 #6-10 even, 16, 17, 26, 28
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