CAHSEE W. UP GEOMTRY GAME PLAN Date9/24/13 Tuesday Section / TopicNotes #19: 2.2 Definitions &...
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Transcript of CAHSEE W. UP GEOMTRY GAME PLAN Date9/24/13 Tuesday Section / TopicNotes #19: 2.2 Definitions &...
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CAHSEE W. UP
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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
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Lesson 2.2Definitions and Biconditional Statements
“The better part of one’s life consists of his friendships.” –Abraham Lincoln
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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.
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Perpendicular lines
Two lines that intersect to form a right angle (90 degrees).
n
m
n mmeans “is perpendicular to”
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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
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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.
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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
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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!
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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!
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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!
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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!
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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
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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!
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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.
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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
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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…
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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.
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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.
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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.
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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.
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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.
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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.