CAHSEE W. UP GEOMTRY GAME PLAN Date9/24/13 Tuesday Section / TopicNotes #19: 2.2 Definitions & Biconditional Statements Lesson GoalSTUDENTS WILL BE ABLE.

CAHSEE W. UP

GEOMTRY GAME PLAN

Date 9/24/13 Tuesday

Section / Topic Notes #19: 2.2 Definitions & Biconditional Statements

Lesson Goal STUDENTS WILL BE ABLE TO RECOGNIZE, ANALYZE AND USE BICONDITIONAL STATEMENTS.

Geometry California Standard 3.0 Students construct and judge the validity of a logical argument and give counterexamples to disprove a statement.

Homework #19 P.76 #18-28 all P. 82 #13-37 EOO

Lesson 2.2Definitions and Biconditional Statements

“The better part of one’s life consists of his friendships.” –Abraham Lincoln

REVIEW: Converse

Switch the hypothesis & conclusion parts of a conditional statement.

Ex: Write the converse of: “If you are a brunette, then you have brown hair.” If you have brown hair, then you are a brunette.

Perpendicular lines

Two lines that intersect to form a right angle (90 degrees).

n

m

n mmeans “is perpendicular to”

Ex: Use definitions to justify your True or False answers.

a.) D, B, & E are collinear.TrueThey are on the same line

b.) TrueThey form right angles

c.) AEB is adjacent to CEDFalseThey do not share a common

side

D E B

A

C

Biconditional statementsStatement that are equivalent to

writing a conditional statement AND its converse◦containing the phrase “if and only if”

◦Conditional: If Rebecca sleeps all morning, then she is sick.

◦Converse: If Rebecca is sick, she sleeps all morning.

◦Biconditional: Rebecca sleeps all morning if and only if she is sick.

Biconditional statements are true sometimes and false sometimes

In order for it to be true, the conditional statement and its converse must both be true◦Then we say the statement is true

“forwards” and “backwards.”

Hints

Rewrite the biconditional statement as a conditional statement and its converse.

◦Two angles are congruent if and only if 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.

I Do!

Rewrite the biconditional statement as a conditional statement and its converse.◦ A ray bisects an angle if and only if it divides

the angle into two congruent angles.

◦ Conditional: If a ray bisects an angle, then it divides the angle into two congruent angles.

◦ Converse: If a ray divides an angle into two congruent angles, then the ray bisects the angle.

We Do!

Rewrite the biconditional statement as a conditional statement and its converse.

Two lines intersect if and only if their intersection is exactly one point.

◦Conditional: If two lines intersect, then they intersect in exactly one point.

◦Converse: If two lines contain exactly one point, then the two lines intersect.

You Do!

Consider the following statement: x2 < 49 if and only if x<7.

Step 1: Is this a biconditional statement? Yes, it contains the phrase “if and only if”

Step 2: Are the conditional and converse true?

Conditional: If x2 < 49, then x<7. True.

Converse: If x<7, then x2 < 49.False. If x = –8, then (-8)2 = 64 which is

not less than 49.

THE BICONDITIONAL IS FALSE

I Do!

We Do!

Consider the following statement:x2 = 4 if and only if x = 2

Step 1: Is this a biconditional statement? Yes, it contains the phrase “if and only if”

Step 2: Are the conditional and converse true? Conditional: If x2 = 4, then x = 2.

True. Converse: If x = , then x2 = 4.

True.

THE BICONDITIONAL IS TRUE

Consider the following statement: y = -3 if and only if y2 = 9.Conditional: If y = -3, then y2 = 9.

True. If y = -3, then (-3)2 = 9.

Converse: If y2 = 9, then y = -3. False. 32 = 9, so y can also be positive

3,

THE BICONDITIONAL IS FALSE

You Do!

TOD!

#1 Is the biconditional statement true or false.

• y = -3 if and only if y2 = 9.

#2 Is the biconditional statement true or false.

• An angle measures 94º if and only if it is obtuse.

TOD!#1 y = -3 if and only if y2 = 9.

Conditional: If y = -3, then y2 = 9. True. If y = -3, then (-3)2 = 9.

Converse: If y2 = 9, then y = -3. False. 32 = 9, so y can also be positive 3,

BICONDITIONAL STATEMENT FALSE

#2 An angle measures 94º if and only if it is obtuse.

Conditional: If an angle measures 94º, then it is obtuse. TRUE.

Converse: If an angle is obtuse, then it measures 94º. FALSE. If an angle is obtuse, then it can measure any degree between 90º and 180º.

BICONDITIONAL STATEMENT FALSE

Give a counterexample that demonstrates that the converse of the statement is false.If an angle measures 94°, then it is obtuse.

If two angles measure 42° and 48°, then they are complementary.

More examples…

Rewrite each of the following statements in “If-then” form as the conditional, converse, inverse, contrapositive, and biconditional.

1) Celebrities have many fans.

Conditional: If you are a celebrity, then you have many fans.

Converse: If you have many fans, then you are a celebrity.

Inverse: If you are not a celebrity, then you do not have many fans.

Contrapositive: If you do not have many fans, then you are not a celebrity.

Biconditional: You are a celebrity if and only if you have many fans.

2) Penguins are birds that cannot fly.

Conditional: If a bird is a penguin, then it cannot fly.

Converse: If a bird cannot fly, then it is a penguin.

Inverse: If a bird is not a penguin then it can fly.

Contrapositive: If a bird can fly, then it is not a penguin.

Bionditional: A bird is a penguin if and only if it cannot fly.

3) Angles that form a linear pair are supplementary.

Conditional: If two angles form a linear pair, then they are supplementary.

Converse: If two angles are supplementary, then they form a linear pair.

Inverse: If two angles do not form a linear pair, then they are not supplementary.

Contrapositive: If two angles are not supplementary, then they do not form a linear pair.

Biconditional: Two angles form a linear pair if and only if they are supplementary.

4) Complementary angles are acute.

Conditional: If two angles are complementary, then they are acute.

Converse: If two angles are acute, then they are complementary.

Inverse: If two angles are not complementary, then they are not acute.

Contrapositive: If two angles are not acute, then they are not complementary.

Biconditional: Two angles are complementary if and only if they are acute.

5) I will go to school on Monday.

Conditional: If I go to school, then it’s Monday.

Converse: If it’s Monday, then I will go to school.

Inverse: If I don’t go to school, then it is not Monday.

Contrapositive: If it is not Monday, then I will not go to school.

Biconditional: I will go to school if and only if it is Monday.