Draw a Logical Conclusion:  If you are a lefty then you struggle to use a can opener.  If you like math then you must be smart.  If you are smart then you are a lefty.

Unit Essential Question: How do you use appropriate reasoning to reach a given conclusion?

Essential Question: What are the forms of the conditional statement?

Conditional Statements  If-Then statements like: If you are not satisfied, then you will get a refund.  Hypothesis –  Conclusion –

Identifying Hypothesis & Conclusion  If you like the Patriots then you are wise.  If it is raining then bring your umbrella.

Writing as a Conditional  Rectangles have four right angles.  Tigers have stripes.

Truth Values  Either True or False True – Every time hypothesis is true, the conclusion is also true. False – Find ONE counterexample where the hypothesis is true, but conclusion is false.

Find Truth Values:  If it is February then there are only 28 days in the month.  If the name of a state starts with NEW then it borders an ocean.

Using a Venn Diagram

Converse  The converse of the conditional switches the hypothesis and conclusion.  Ex: If two lines intersect to form right angles, then they are perpendicular.  Converse: If two lines are perpendicular, then they intersect to form right angles.

Truth Value of Converses

Truth Values of Conditionals AND Converses:  If two lines do not intersect, then they are parallel.  If x = 2, then | x | = 2

Forms of the Conditional

Write your own  Write your own conditional statement.  Have your neighbor write the converse.

Hypothesis & Conclusion

What are the (other) forms of a Conditional Statement?

Biconditional  Biconditional – A conditional where both the Conditional & Converse are true. P ↔ q Ex: A quadrilateral is a rectangle if and only if it has four right angles.

Writing a Biconditional  If two segments have the same length then they are congruent.

Is the Biconditional true?  If three points lie on the same line, then they are collinear.

Two Conditionals in Biconditional:  You will pass this geometry course if and only if you are successful with your homework.

Split into Two Conditionals  A number is divisible by 3 if and only if the sum of its digits is divisible by 3.

Definitions & Polyglobs

A Good Definition

Perpendicular Lines  Show that the definition of Perpendicular Lines is a good definition:  Perpendicular Lines intersect to form a right angle.

Good Definitions?  An airplane is a vehicle that flies.  A triangle has sharp corners.  A square has four equal sides.

What are the (other) forms of a Conditional Statement?

Negation  The Negation of a statement has the opposite truth value.  Negate: Philadelphia is the capitol of Pennsylvania.

Negate the following Statements  Today is not Tuesday.  <ABC is obtuse.  Lines m and n are perpendicular.

Inverse & Contrapositive  An Inverse negates the hypothesis and conclusion of a conditional.  The Contrapositive switches and negates the hypothesis and conclusion of a conditional.

Forms of Conditional  Negation: ~P  Conditonal: P→Q  Converse: Q→P  Inverse: ~P→~Q  Contrapositive: ~Q→~P  Biconditional (if and only if): P↔Q

Forms of Conditional – Truth  Conditional: If a figure is a square then it is a rectangle:  Converse:  Inverse:  Contrapositive:

Indirect Reasoning  In Indirect Reasoning all possibilities are considered and all but one are shown false.

Indirect Proof:  Prove that two acute angles cannot be supplementary.  IF  THEN  BUT  SO

Contradictions  A Contradiction occurs when two (or more) statements cannot be true simultaneously.

Indirect Proof:  Prove that a triangle cannot have two obtuse angles.  IF  THEN  BUT  SO

What do you think?

Essential Question: How do you make conclusions using Deductive Reasoning?

Deductive Reasoning  Deductive Reasoning is the process of reasoning logically from given statements to a given conclusion. If the given statements are true, deductive reasoning produces a true conclusion.

Law of Detachment  What does this mean?

Draw a Conclusion  A midpoint divides a segment into two congruent segments.  M is the midpoint of AB.

What do you think?

Law of Syllogism

Using Law of Syllogism

Detachment & Syllogism

Essential Question: How do you justify the steps in solving an Algebraic Equation?

Solve  Steps:  4x – 7 = 29  Justifications:

Properties of Equality

Which Properties? AlgebraProperties  17 = 3x – 4 

Distributive Property  3(x – 7) =  15x 3 – 25x 2 =

Justifying your Steps

Fill in Blanks

Solve & Justify

Properties of Congruence

Justifying Steps  What does it mean to Justify your steps?

Reviewing Important Concepts

How do you prove a Statement?

PROVING a Theorem

Solve & Justify

Congruent Supplements

Can you Prove these Theorems?