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Lesson 2-2
Properties from Algebra
(page 37)
Essential Question
Can you justify the conclusion
of a conditional statement?
Properties from Algebra
• “What are you bringing to the table?”
• Do you remember your properties from
algebra?
• You must keep your properties,
definitions, postulates, and theorems in
your “bag of tricks.”
Properties of Equality (=)
Addition Property:
If a = b and c = d, then a + c = b + d
Subtraction Property:
If a = b and c = d, then a - c = b - d
Multiplication Property:
If a = b, then ca = cb
Division Property:
If a = b and c ≠ 0, then a/c = b/c
Substitution Property:
If a = b, then either a or b may be
substituted for the other in any
equation or inequality.
… Properties of Equality (=)
Reflexive Property:
a=a
Symmetric Property:
If a = b, then b = a
Transitive Property:
If a = b and b = c, then a = c
Distributive Property:
a (b + c) = ab + ac
Properties of Congruence (
)
Reflexive Property:
Symmetric Property:
DE @ DE or ÐD @ ÐD
If DE @ FG , then FG @ DE
If ÐD @ ÐE , then ÐE @ ÐD
Transitive Property:
If DE @ FG and FG @ JK, then DE @ JK
If ÐD @ ÐE and ÐE @ ÐF, then ÐD @ ÐF
Statement
a.
If AB = CD and BC = BC,
then AB + BC = CD + BC.
Justification
Addition Property of Equality
Statement
b.
If 2 + YZ = 8, then YZ = 6.
Justification
Subtraction Property of Equality
Statement
c.
If 2 m∠1=72º, then m∠1=36º.
Justification
Division Property of Equality
Statement
d.
If Pt. B is between A and C,
then AB + BC = AC .
Justification
Segment Addition Postulate
Statement
e.
If m∠A = m∠X & m∠X = m∠B,
then m∠A = m∠B .
Justification
Transitive Property of Equality
Statement
f.
If AB @ CD and CD @ EF, then AB@ EF.
Justification
Transitive Property of Congruence
Statement
g.
If ÐA @ ÐB, then ÐB@ ÐA.
Justification
Symmetric Property of Congruence
Statement
h.
If mÐK = 90º, then ÐK is a right angle.
Justification
Definition of Right Angle
or Def. Rt. ∠
Solve for “x” and supply reasons for each step.
Steps
1. 5 ( 3 x - 4 ) = 5 x
Reasons
_____________________________________________
Solve for “x” and supply reasons for each step.
Steps
Reasons
1. 5 ( 3 x - 4 ) = 5 x
_____________________________________________
2.
_____________________________________________
15 x - 20 = 5 x
Solve for “x” and supply reasons for each step.
Steps
Reasons
1. 5 ( 3 x - 4 ) = 5 x
_____________________________________________
2.
15 x - 20 = 5 x
_____________________________________________
3.
10 x - 20 = 0
_____________________________________________
Solve for “x” and supply reasons for each step.
Steps
Reasons
1. 5 ( 3 x - 4 ) = 5 x
_____________________________________________
2.
15 x - 20 = 5 x
_____________________________________________
3.
10 x - 20 = 0
_____________________________________________
4.
10 x = 20
_____________________________________________
Solve for “x” and supply reasons for each step.
Steps
Reasons
1. 5 ( 3 x - 4 ) = 5 x
_____________________________________________
2.
15 x - 20 = 5 x
_____________________________________________
3.
10 x - 20 = 0
_____________________________________________
4.
5.
10 x = 20
x=2
Given Problem
Distributive Property
Subtraction Prop. of =
Addition Prop. of =
_____________________________________________
Division Prop. of =
_____________________________________________
Complete the proof.
D
E
Given: m∠1 = m∠3
Prove: m∠ABE = m∠DBC
Statements
m∠1 = m∠3
1. ___________________
1
A
2
B
3
C
Reasons
Given
m∠2 = m∠2
2. ___________________
Reflexive
Property of =
___________________
3. m∠1
___________________
+ m∠2 = m∠3 + m∠2
Addition Prop. of =
___________________
4. m∠ABE = _____________
m∠1 + m∠2
___________________
Angle Addition Post.
m∠DBC = _____________
m∠3 + m∠2
5. ___________________
m∠ABE = m∠DBC
Substitution Prop.
___________________
Complete the proof.
Given: KP = ST ; PR = TV
Prove: KR = SV
Statements
1. ___________________
KP = ST ; PR = TV
2. ___________________
KP + PR = ST + TV
3. KP + PR = _____________
KR
K
P
R
S
T
V
Reasons
Given
Addition Prop. of =
___________________
___________________
Segment Add. Post.
ST + TV = _____________
SV
4. ___________________
KR = SV
Substitution Prop.
___________________
Assignment
Written Exercises on pages 41 & 42
GRADED: 1 to 9 odd numbers
Be sure to write out the entire proof!
Can you justify the conclusion
of a conditional statement?