Warm-Up

1. mAOC = ?

2. mAOD = ?

3. mDOB = ?

4. Name a right angle.

Review of Logic Statements

The conditional: If you work then you get money.

Underline the hypothesis. Double underline the conclusion.

Write each of these formats and show the PQ formula. Negate the hypothesis. Converse Inverse Contrapositive Biconditional

One More Time

Conditional:If you are polite, then others smile.

ConverseInverseContrapositiveBiconditional

Quiz: Logic StatementsFill in the missing word or symbols using the conditional A→B

1)_________ 2) A↔B 3) _________ 4) ~A→~B 5)__________

1) Converse 2) _________ 3)

Contrapositive 4)___________ 5) Negation of

the Conclusion.

Reasoning & ProofChapter 2

2.3 – Deductive Reasoning2.4 – Reasoning in Algebra

Remember…

Inductive reasoning is based on patterns you observe.

Today we will learn about deductive reasoning.

Deductive Reasoning Deductive reasoning (logical reasoning)

is the process of reasoning logically from given statements to a conclusion.

If the given statements are true, deductive reasoning produces a true conclusion.

Ex.1: Deductive Reasoning

An auto mechanic knows that if a car has a dead battery, the car will not start. A mechanic begins work on a car and finds the battery is dead. What conclusion can they make?

The car will not start.

Law of Detachment If a conditional is true and its hypothesis is

true, then its conclusion is true.

If pq is true and p is true, then q is true.

If it is cold out side, then I will wear a jacket.

Ex.2: Using Law of DetachmentFor the given statements, use the Law of Detachment to draw a conclusion.

a) If a figure is a rectangle, then it has two pairs of parallel sides.Figure ABCD is a rectangle.

b) If a road is icy, then driving conditions are hazardous.Driving conditions are hazardous.Not possible; you do not know

that the hypothesis is true.

Figure ABCD has two pairs of parallel sides.

Detachment

Common Sense!

Duhhhh!

Law of Syllogism Allows you to state a conclusion from two true

conditional statements when the conclusion of one statement is the hypothesis of the other statement.

If it is lunch time, then I am hungry. If I am hungry, then I want pizza!

Law of Syllogism“Slide”

See the if then format Knock out the phrase that is

repeated Read the new small if then

statement

Remember

The answer for a syllogism is always an if then statement NOT a phrase.

Ex.3a: Using the Law of SyllogismUse the Law of Syllogism to draw a conclusion.

a) If a river is more than 4000 mi long, then it is longer than the Amazon.

b) If a river is longer than the Amazon, then it is the longest river in the world.

The Nile is 4132 mi long.

If a river is more than 4000 mi long, then it is the longest river in the world.

#3b)

If the circus is in town, then there are tents at the fairground.

If there are tents at the fairground, then Paul is working as a night watchmen.

The circus is in town.

Properties of EqualityIf a = b then…

Addition Property

Subtraction Property

Multiplication Property

Division Property

a + c = b + c

a – c = b – c

a * c = b * c

c 0, then a/c = b/c

It’s easy asR-S-T 1-2-3

Reflexive 1. A A

Symmetric 2. If A B, then B A

Transitive 3. If A B and B C, then A C

Ex.4a: Justifying Steps in Solving an Equation

5(x + 3) = -5 5x + 15 = -5 5x = -20 x = -4

Ex.4b: Solve for x. Justify each step.

KM = 35.

KL + LM = KM

In-Class Practice

Textbook - p. 93 #29 G E

(9x – 2)º móGFI = 128

4xº

F I

a) Solve for x. Justify each step. b) Find m óEFI.

Pg. 80 #47In a band, Amy, Bob, and Carla are the drummer, guitarist, and keyboard player. Use the clues to find the instrument that each one plays.

Carla & the drummer wear different colored shirts. The keyboard player is older than Bob. Amy, the youngest band member, lives next door to

the guitarist.

Pg. 80 #49Alan, Ben, and Cal are seated as shown with their eyes closed. Diane places a hat on each of their heads from a box they know contains 3 red and 2 blue hats. They open their eyes and look forward.

Alan says,“I cannot deduce what

color hat I’m wearing.”Hearing that, Ben says, “I

cannot deduce what color I’m wearing, either.”

Cal then says, “I know what color I’m wearing!”

How does Cal know the color of his hat?