Geometry

1
Geometry

• Chapter 2 Logic and Reasoning

2
2.4 Using Proof in Algebra

• Objectives
• Complete an algebraic proof

3
What you know

• Segment Addition
• Definition of Midpoint
• Angle Addition
• Law of Syllogism Law of Detachment

4
Vocabulary

• Deductive Reasoning
• Two column proof
• flow proof
• reflexive property
• symmetric property
• transitive property
• addition/subtraction property of equality
• multiplication and division property of equality

• substitution
• distributive
• algebraic proof
• Definition of Congruence
• Angle Addition Property
• Segment Addition Property
• Definition of Midpoint

5
Vocabulary Parts of Proof

• Deductive Reasoning
• Two column proof
• flow proof

6
Vocabulary Properties of Equality
Reflexive
55
BC?BC
If 5x Then x5
If AB ? BD Then BD ? AB
Symmetric
If x 3y and yz Then x3z
If AB ? BD and BD ? AD Then AB ? AD
Transitive
If x 3 5 Then x 2
Addition and Subtraction Property of Equality
If ?1 ?2 ?2 ?3 Then ?1 ?3
Multiplication and Division Property of Equality
If 2x 6 Then x 3
If 2AC 10 Then AC 5
Substitution Property of Equality
If ?1?2180 and ?1?3180 Then ?1?2?1?3
If x y 3 and y 5 Then x 5 3
If 2(x 3) Then 2x 6
Distributive Property
IF 2(AC CD) Then 2AC 2CD
7
Vocabulary Other Properties
Definition of Congruence
If m?1m?2 Then ?1??2
If AB ? CD Then AB CD
A
Angle Addition Property
m?ABCm?CBDm?ABD
C
B
D
E
D
F
DEEF DF
Segment Addition Property
If M is the midpoint of PQ Then PM ? MQ
Definition of Midpoint
M
P
Q
8
Vocabulary Other Properties
Definition of Congruence
Angle Addition Property
Segment Addition Property
Definition of Midpoint
9
Setting up an Algebraic Proof
Prove If 2(x 3) 12 then x3
Given
2(x3)12
x3
Prove
Reasons
Statements
1.
2(x3)12
1. Given
2. 2x 6 12
2. Distribution
3. 2x 6
3. Add/ Sub Property of Equality
4. x 3
4. Mult/Div Property of Equality
10
Examples
A
Z
Given AB?XY BC?YZ
Y
X
B
C
Prove AC?XZ
Statements
Reasons
1. AB?XY, BC?YZ
1.
Given
2. AB XY, BC YZ
2.
Definition of Congruence
3. AB BC XY YZ
3.
Addition Property of Equality
4.
Segment Addition Postulate
4. AB BC AC XY YZ XZ
5. AC XZ
5.
Substitution Property of Equality
6. AC?XZ
6.
Definition of Congruence
11
Homework

• HW Page 96, 22 29