1. Warm Up

2. Warm Up Answers

3. Monty Python’s Crazy Logic (click on the image to view video)
2.5 Algebraic Proof
Monty Python’s Crazy Logic
(click on the image to view video)

4. 2.5 Algebraic Proof
Objectives: Review properties of equality and use them to write algebraic proofs. Identify properties of equality and congruence. Proof: An argument that uses logic, definitions, properties, and previously proven statements to show that a conclusion is true.

5. Section 2-5: Reasoning in Algebra Standard: apply reflective, transitive, or symmetric properties of equality or congruence
Objectives:
Connect reasoning in algebra and geometry
Justify steps in deductive reasoning
In geometry
postulates, definitions, & properties are accepted as true
(refer to page 842 for a complete list of postulates)
you use deductive reasoning to prove other statements
We will look at some basic properties used to justify statements…..
….. which leads to writing proofs.

6. Properties of Equality
Page 113
Addition Property of Equality
If a = b, then a + c = b + c
Add same amount to both sides of an equation.
Subtraction Property of Equality
If a = b, then a - c = b - c
Subtract same amount to both sides of an equation.
Multiplication Property of Equality
If a = b, then a ∙ c = b ∙ c
Multiply both sides of an equation by the same amount.
Division Property of Equality
If a = b and c  0, then
Divide both sides of an equation by the same amount.

7. Properties of Equality (cont)
Reflective Property of Equality
a = a Ex: 5 = 5
Symmetric Property of Equality
If a = b, then b = a Ex: 3 = and = 3 are the same.
Transitive Property of Equality
If a = b and b = c, then a = c.
EX: If = 7 and = 7, then =
Substitution Property of Equality
If a = b , then b can replace a in any expression.
Ex: a = 3; If a = b, then 3 = 3.
Distributive Property
a(b + c) = ab + ac Ex: 3(x + 3) = 3x + 9

8. 2.5 Properties of Equality Table on page #113
The Distributive Property states that
a(b + c) = ab + ac.
Remember!

9. Properties of Congruence
The Reflective, Symmetric, and Transitive Properties of Equality have corresponding properties of congruence that can be used to justify statements.
Reflective Property of Congruence
AB  AB
A  A
Symmetric Property of Congruence
If AB  CD, then CD  AB.
If A  B, then B  A
Transitive Property of Congruence
If AB  CD and AB  EF, then CD  EF.
If A  B and B  C, then A  C.

10. 2.5 Properties of Congruence Table on page #114

11. What’s the Difference between equality and congruence?
A B
AB represents the length AB, so you can think of AB as a variable representing a number.
Helpful Hint

12. Geometric objects (figures / drawings) can be congruent to each other.
Congruence
Equality
Geometric objects (figures / drawings) can be congruent to each other.
Measurements (numbers)can be equal to each other.
Statements use symbol
Statements use = symbol
Numbers are equal (=) and
figures are congruent ().
Remember!

13. 2.5 Application
Write a justification for each step. NO = NM + MO Segment Addition Post. 4x – 4 = 2x + (3x – 9) Substitution Property of Equality 4x – 4 = 5x – 9 Simplify. –4 = x – 9 Subtraction Property of Equality 5 = x Addition Property of Equality

14. The basic format of a two column proof: Page 115
Given - facts you are given to use.
STARTING POINT
Prove – conclusion you need to reach.
ENDING POINT

15. Proof Example: Problem 3 page 116
This is how you plan to get from the given to the prove.
This is given
This is what you are asked to prove
Reasons

16. Application Statement Reason AB + BC = AC 2y + 3y – 9 = 21 5y – 9 = 21
PROVE: y = 6
GIVEN:
Statement
Reason
AB + BC = AC
2y + 3y – 9 = 21
5y – 9 = 21
5y = 30
y = 6
Segment addition postulate
Substitution
Combine like terms
Addition Property (add 9 to both sides)
Division property (divide both sides by 5)

17. Using Properties to Justify Steps in Solving Equations
Algebra: Prove x = 43 and justify each step.
Given: m AOC = 139
Prove : x = 43
Statement
Reasons
m AOC = 139
Given
M AOB + m BOC = m AOC
Angle Addition Postulate
x x = 139
Substitution Property
Simplify or combine like terms
3x + 10 = 139
3x = 129
Subtraction Property of Equality
x = 43
Division Property of Equality

18. Using Properties to Justify Steps in Solving Equations
Prove x = 20 and justify each step.
Given: LM bisects KLN
Prove: x = 20
Statement
Reasons
LM bisects KLN
Given
MLN = KLM
4x = 2x + 40
2x = 40
x = 20
Def of Angle Bisector
Substitution Property
Subtraction Property of Equality
Division Property of Equality

19. Using Properties to Justify Steps in Solving Equations
Now you try
Solve for y and justify each step
Given: AC = 21
Prove : y = 6
Statement
Reasons
AC = 21
Given
AB + BC = AC
Segment Addition Postulate
2y + 3y - 9 = 21
Substitution Property
Simplify
5y – 9 = 21
5y = 30
Addition Property of Equality
y = 6
Division Property of Equality
Find AB and BC by substituting y = 6 into the expressions.

20. Using Properties of Equality and Congruence
Name the property of congruence or
equality the justifies each statement.
a. K  K
Reflective Property of Congruence
b. If 2x – 8 = 10, then 2x = 18
Addition Property of Equality
c. If RS  TW and
TW  PQ,
then RS  PQ.
Transitive Property of Congruence
d. If m A = m B, then
m B = m A
Symmetric Property of Equality

21. Use what you know about transitive properties to complete the following:
The Transitive Property of Falling Dominoes:
If domino A causes domino B to fall, and domino B causes domino C to fall, then domino A causes domino _______ to fall. C

22. HOMEWORK
COMPLETE 2-5 PACKET
DUE THURSDAY NOV 1