Geometry - Mr Hickman's Class 2020-2021...September 7, 2016 GEOMETRY 2.1 CONDITIONAL STATEMENTS. Write the following biconditional statement as a conditional statement and its converse

GEOMETRY 2.1 Conditional Statements

September 7, 2016 2.1 CONDITIONAL STATEMENTS

and distance.

ESSENTIAL QUESTION

When is a conditional statement true or false?


WHAT YOU WILL LEARN

oWrite conditional statements.

oUse definitions written as conditional statements.

oWrite biconditional statements.



CONDITIONAL

A type of logical statement that has two parts, a hypothesis and a conclusion.

A conditional can be written in IF-THEN form.


SHORTHAND

If HYPOTHESIS, then CONCLUSION.

If P, then Q.

In the study of logic, P’s and Q’s are universally accepted to represent hypothesis and conclusion.


EXAMPLE 1

If I study hard, then I will get good grades.

HYPOTHESIS

I study hard

CONCLUSION

I will get good grades.


CAN YOU IDENTIFY THE HYPOTHESIS AND CONCLUSION?

If today is Monday, then tomorrow is Tuesday.

Hypothesis: today is Monday

Conclusion: tomorrow is Tuesday.

Note: IF is NOT part of the hypothesis, and THEN is NOT part of the conclusion.


YOUR TURN

Underline the hypothesis and circle the conclusion.


1. If the weather is warm, then we should go swimming.

2. If you want good service, then take your car to Joe’s Service Center.

REWRITING STATEMENTS.

oUse common sense.

oDon’t over analyze it.

oMake sure the sentence is grammatically correct.


The hypothesis always follows “IF.”

No “if?” The first part is usually the hypothesis.

Make your English teacher proud!Does it sound right?

EXAMPLE 2A

Rewrite the following statement in if-then form:

All birds have feathers.


What is the hypothesis?

What is the conclusion? have feathers

All birds

If-then form?

If an animal is a bird, then it has feathers.

EXAMPLE 2B


You are in Texas if you are in Houston.



What is the conclusion? You are in Texas

You are in Houston

If-then form?

If you are in Houston, then you are in

Texas.

EXAMPLE 2C


An even number is divisible by 2.



What is the conclusion? Divisible by 2.

An even number

If-then form?

If a number is even, then it is divisible by 2.

YOUR TURN

Rewrite the conditional statement in if-then form.

If yesterday was Sunday, then today is

Monday.

If an object measures 12 inches, then it is one

foot long.


3. Today is Monday if yesterday was Sunday.

4. An object that measures 12 inches is one foot long.

NEGATION

The negative of the original statement. Examples:

I am happy.

I am not happy.

mC = 30°.

mC 30°.

A, B and C are on the same line.

A, B and C are not on the same line.


NEGATION


EXAMPLE 3

Write the negation of each statement.

a. The ball is red.

The ball is not red.

b. The cat is not black.

The cat is black.

c. The car is white.

The car is not white.



RELATED CONDITIONAL STATEMENTS

Looking at the conditional statement: If p, then q.

There are three similar statements we can make.


o Converseo Inverseo Contrapositive

CONVERSE

The converse of a statement is formed by

switching the hypothesis and the conclusion.

If you play drums, then you are in the band.

Conditional:

Converse:

If you are in the band, then you play drums.


If Q, then P.

EXAMPLE 4

Write the converse of the statement below.

Answer:

If you play on the tennis team, then you like tennis.


If you like tennis, then you play on the tennis team.

INVERSE

The inverse is formed by taking the negation

of the hypothesis and of the conclusion.

Conditional:

If x = 3, then 2x = 6.

Inverse:

If x 3, then 2x 6.


If not P, then not Q.

EXAMPLE 5

Write the inverse of the statement below.

Answer:

If today is not Monday, then tomorrow is not Tuesday.


If today is Monday, then tomorrow is Tuesday.

CONTRAPOSITIVE

The contrapositive is formed by switching and negating

the hypothesis and the conclusion.

(Take the inverse of the converse, or, the converse of the

inverse.)

Conditional:

If I am in 10th grade, then I am a sophomore.

Contrapositive:

If I am not a sophomore, then I am not in 10th grade.


If not Q, then not P.

EXAMPLE 6

Write the contrapositive of the statement below.


If x is odd, then x + 1 is even.

x + 1 is not evenNegateNegate

x is not odd

If x+1 is not even, then x is not odd.

LOGICAL STATEMENTS

If I live in Mesa, then I live in Arizona.

Converse: (switch hypothesis and conclusion)

If I live in Arizona, then I live in Mesa.

Inverse: (negate hypothesis and conclusion)

If I don’t live in Mesa, then I don’t live in Arizona.

Contrapositive: (switch and negate both)

If I don’t live in Arizona, then I don’t live in Mesa.


YOUR TURN. WRITE THE CONVERSE, INVERSE, AND CONTRAPOSITIVE.

If mA = 20, then A is acute.


If A is acute, then mA = 20.


If mA 20, then A is not acute.


If A is not acute, then mA 20.


REVIEW: LOGICAL STATEMENTS


Conditional: If P, then Q.

Converse: If Q, then P.

Inverse: If not P, then not Q.

Contrapositive: If not Q, then not P.

DEFINITION: PERPENDICULAR LINES


Two lines that intersect to form a right angle.

m

n

Notation:

m n

USING DEFINITIONS

You can write a definition as a conditional statement in if-then form. Let’s look at an example:

The conditional statement would be:

The converse statement also ends up being true:


Perpendicular Lines: two lines that intersect to form a right angle.

If two lines are perpendicular, then they intersect to form a

right angle.

If two lines intersect to form a right angle, then they are

perpendicular lines.

DAY 2 2.1 Conditional Statements

TRUTH VALUES

•A conditional is either True or False.

•To show that it is true, you must have an argument (a proof) that it is true in all cases.

•To show that it is false, you need to provide at least one counterexample.


EXAMPLE 7True or false? If false provide a counter example.

If x2= 9, then x = 3.

FALSE!Counterexample: x could be –3.


EXAMPLE 8

If x = 10, then x + 4 = 14.

True! Proof:

x = 10

x + 4 = 10 + 4

x + 4 = 14


EQUIVALENT STATEMENTS

When two statements are both true or both false, they are called equivalent statements.

A conditional statement is always equivalent to its contrapositive.

The inverse and converse are also equivalent.


EQUIVALENT STATEMENTS

Original:

If mA = 20, then A is acute.


If A is acute, then mA = 20.


If mA 20, then A is not acute.


If A is not acute, then mA 20.

TRUE

False

False

TRUE


EXAMPLE 9

Statement: If x = 5, then x2 = 25. TRUE

Contrapositive: If x2 25, then x 5. TRUE

Converse: If x2 = 25, then x = 5. FALSE – could be –5.

Inverse: If x 5, then x2 25. FALSE


JUSTIFYING STATEMENTS

September 7, 2016 GEOMETRY 2.1 CONDITIONAL STATEMENTS

In math, deciding if a statement is true or false demands that you can justify your answers. “Just because”, or, “It looks like it” are not sufficient.

Justification must come in the form of Postulates, Definitions, or Theorems.

EXAMPLE 10


A

XD B

C

Statement

Truth Value

Reason

D, X, and B are collinear.

TRUE

Definition of collinear points.

EXAMPLE 11


A

XD B

C

Statement

Truth Value

Reason

AC DB

TRUE

Definition of Perpendicular lines

Def lines

EXAMPLE 12


A

XD B

C

Statement

Truth Value

Reason

CXB is adjacent to BXA

TRUE

Def. of adjacent angles

Def. of adj. s

EXAMPLE 13


A

XD B

C

Statement

Truth Value

Reason

DXA and CXB are adjacent angles.

FALSE

There is not a common side. (Or, they are vertical angles.)

VERY IMPORTANT!


In doing proofs, you must be able to justify every statement with a valid reason. To be able to do this you must know every definition, postulate and theorem. Being able to look them up is no substitute for memorization.

YOUR TURN


A

F

BE

D

C

G

H

YOUR TURN


A

F

BE

D

C

G

H

False (they are not collinear)

True (add to 180 )

True (post. 8)

False (no rt. mark)

YOUR TURN


A

F

BE

D

C

G

H

True (def. lines)

False (they are supplementary)

True (half of 180 is 90 -- a right )

BICONDITIONALS


If 2 s are complementary, then their sum is 90°. True

Converse

If the sum of 2 s is 90°, then they are complementary.True

When a conditional statement and its converse are both TRUE,

they can be written as a single biconditional statement. Let’s look

at an example:

Conditional

Biconditional

2 s are complementary if and only if their sum is 90°.

BICONDITIONALS (Continued)


Written with p’s and q’s a biconditional looks like this:

p if and only if q.

p iff q. or

Iff means “if and only if”.

PUTTING IT ALL TOGETHER


Statements In words In symbols

Conditional If p, then q 𝑝 → 𝑞

Converse If q, then p 𝑞 → 𝑝

Inverse If not p, then not q ~𝑝 → ~𝑞

Contrapostive If not q, then not p ~𝑞 → ~𝑝

Biconditional p if and only if q 𝑝 ↔ 𝑞

EXAMPLE 14

September 7, 2016 2.3 DEDUCTIVE REASONING 51

Let P be the statement: “x = 3”

Let Q be the statement: “2x = 6”

Write:

P Q

Q P

P Q

If x = 3, then 2x = 6.

If 2x = 6, then x = 3.

x = 3 if and only if 2x = 6.

or 2x = 6 iff x = 3.

DEFINITIONS


ALL definitions are biconditionals.

Example: Definition of Congruent Angles

Two angles are congruent iff they have the same measure.

Conditional: If two angles are congruent, then they have the same measure.

Converse: If two angles have the same measure, then they are congruent.

TRUTH VALUES OF BICONDITIONALS


A biconditional is TRUE if both the conditional and the converse are true.

A biconditional is FALSE if either the conditional or the converse is false, or both are false.

EXAMPLE 15


Biconditional

x = 5 iff x2 = 25.

Conditional

If x = 5, then x2 = 25.

Converse

If x2 = 25, then x = 5.

true

False!

False!

True or False?

True or False?

True or False?

YOUR TURN


Write the following biconditional statement as a conditional statement and its converse.

An angle is obtuse iff it measures between 90 and 180.

AnswerConditional: If an angle is obtuse, then it measures between 90 and 180.Converse: If an angle measures between 90 and 180, then it is obtuse.

WHY IS THIS IMPORTANT?

Geometry is stated in rules of logic.

We use logic to prove things.

It teaches us to think clearly and without error.

It impresses girl friends (or boy friends).

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Geometry - Mr Hickman's Class 2020-2021...September 7, 2016 GEOMETRY 2.1 CONDITIONAL STATEMENTS. Write the following biconditional statement as a conditional statement and its converse