2.1 Algebraic and Segment Proofs.notebook
September 30, 2015
Algebraic Proofs and Proofs about Line Segments
Proof: a logical argument that uses a sequence of statements to prove a conjecture.
Conjecture: Theorem: Postulate: an unproven statement that is based on observations; can be true or false
conjecture proven true
a statement accepted as true without proof
Oct 13­8:15 AM
1
2.1 Algebraic and Segment Proofs.notebook
September 30, 2015
Properties of Equality
Addition Property of Equality
Subtraction Property of Equality
Multiplication Property of Equality
Division Property of Equality
Distributive Property of Equality
Substitution Property of Equality
If a = b, then a can be substituted for b in any expression.
Oct 1­9:47 AM
2
2.1 Algebraic and Segment Proofs.notebook
Addition Prop (=)
Multiplication Prop (=)
September 30, 2015
Subtraction Prop (=)
Division Prop (=)
Substitution Prop (=)
Distributive Prop (=)
Algebraic Proofs
1.) Given: Prove: Reasons
Statements
Oct 9­4:43 PM
3
2.1 Algebraic and Segment Proofs.notebook
Addition Prop (=)
Multiplication Prop (=)
2.) Given: September 30, 2015
Subtraction Prop (=)
Division Prop (=)
Substitution Prop (=)
Distributive Prop (=)
Distributive Prop (=)
Prove: Reasons
Statements
Oct 9­4:43 PM
4
2.1 Algebraic and Segment Proofs.notebook
September 30, 2015
Properties of Equality/Congruence
Reflexive Property
Equality
Congruence
Oct 13­8:53 AM
5
2.1 Algebraic and Segment Proofs.notebook
September 30, 2015
Properties of Equality/Congruence
Symmetric Property
Equality
Congruence
Oct 1­10:11 AM
6
2.1 Algebraic and Segment Proofs.notebook
September 30, 2015
Properties of Equality/Congruence
Transitive Property
Congruence
Equality
Oct 1­10:14 AM
7
2.1 Algebraic and Segment Proofs.notebook
September 30, 2015
Important Concepts for Segment Proofs
Definition of Congruence
Segment Addition Postulate
Definition of Midpoint
Oct 1­10:18 AM
8
2.1 Algebraic and Segment Proofs.notebook
Given
Definition September 30, 2015
=/
Symmetric =/
Addition Prop (=)
Transitive
Division Prop (=)
Reflexive Segment Addition Postulate
Substitution Prop (=)
=/
Simplify
Distributive Prop (=)
def =/
Definition Midpoint
Multiplication Prop (=)
Subtraction Prop (=)
1.) Given: AC = AB + AB
Prove: AB = BC
B
A
Statements
1.) AC = AB + AB
2.) AC = AB + BC
3.) AB + AB = AB + BC
4.) AB = BC
C
Reasons
Given
Segment Addition Postulate
Substitution Prop (=)
Transitive
=/
Substitution Prop (=)
Oct 9­4:51 PM
9
2.1 Algebraic and Segment Proofs.notebook
Given
Definition September 30, 2015
Reflexive Segment Addition Postulate
Addition Prop (=)
=/
Symmetric =/
Substitution Prop (=)
Transitive
=/
Division Prop (=)
Simplify
Distributive Prop (=)
def =/
Definition Midpoint
2.) Given: PR = QS
Prove: PQ = RS
Multiplication Prop (=)
Subtraction Prop (=)
P
Q
R
S
Statements
1.) PR = QS
Reasons
Given
2.) PQ +QR = PR
Segment Addition Postulate
3.) QR + RS = QS
Segment Addition Postulate
4.) PQ + QR = QR + RS
5.) PQ = RS
Substitution Prop (=)
Subtraction Prop (=)
Oct 9­4:53 PM
10
2.1 Algebraic and Segment Proofs.notebook
Given
Definition September 30, 2015
Reflexive Segment Addition Postulate
Addition Prop (=)
=/
Symmetric =/
Substitution Prop (=)
Transitive
=/
Division Prop (=)
Simplify
Distributive Prop (=)
Multiplication Prop (=)
def =/
Definition Midpoint
Subtraction Prop (=)
3.) Given: K is between J and L
JK = 6 and KL = 10
J
K
L
Prove: JL = 16
Statements
1.) K is between J and L
2.) JL = JK + KL
3.) JK =6 and KL = 10
4.) JL = 6 + 10
5.) JL = 16
Reasons
Given
Segment Addition Postulate
Given
Substitution Prop (=)
Simplify
Oct 9­4:57 PM
11
2.1 Algebraic and Segment Proofs.notebook
Given
Definition Reflexive September 30, 2015
Addition Prop (=)
=/
Symmetric =/
Substitution Prop (=)
Transitive
=/
Division Prop (=)
Segment Addition Postulate
Simplify
Distributive Prop (=)
def =/
Definition Midpoint
Multiplication Prop (=)
Subtraction Prop (=)
D
4.) Given: AB = BD
BD = BC
Prove: B is the midpoint of A
Statements
1.) AB = BD
BD = BC
B
C
Reasons
Given
Transitive
=/
2.) AB = BC
3.) 4.) B is the midpoint of def =/
Definition Midpoint
Oct 9­4:59 PM
12
2.1 Algebraic and Segment Proofs.notebook
Given
Definition Reflexive Segment Addition Postulate
September 30, 2015
Addition Prop (=)
=/
Symmetric =/
Substitution Prop (=)
Transitive
=/
Division Prop (=)
Simplify
Distributive Prop (=)
Multiplication Prop (=)
def =/
Definition Midpoint
5.) Given: AB = 5 and BC = 6
Prove: AC = 11
Statements
Subtraction Prop (=)
A
B
C
Reasons
1.) AB = 5 and BC = 6
2.) AC = AB + BC
3.) AC = 5 + 6
4.) AC = 11
Oct 9­4:59 PM
13
2.1 Algebraic and Segment Proofs.notebook
Given
Definition Reflexive Segment Addition Postulate
September 30, 2015
Addition Prop (=)
=/
Symmetric =/
Substitution Prop (=)
Transitive
=/
Division Prop (=)
Simplify
Distributive Prop (=)
def =/
Definition Midpoint
6.) Given: Prove: Multiplication Prop (=)
Subtraction Prop (=)
P
Q
R
S
Statements
Reasons
1.) 2.) PQ = RS
3.) RS = PQ
4.) Oct 14­2:32 PM
14
2.1 Algebraic and Segment Proofs.notebook
Given
Definition Reflexive Segment Addition Postulate
September 30, 2015
Addition Prop (=)
=/
Symmetric =/
Substitution Prop (=)
Transitive
=/
Division Prop (=)
Simplify
Distributive Prop (=)
def =/
Definition Midpoint
Multiplication Prop (=)
Subtraction Prop (=)
7.) Given: Prove: AC = AB + DE
Statements
A
B
D
E
C
Reasons
1.) 2.) BC = DE
3.) AC = AB + BC
4.) AC = AB + DE
Oct 17­10:07 AM
15
2.1 Algebraic and Segment Proofs.notebook
September 30, 2015
Oct 9­5:03 PM
16