lesson 2.1 conditional statements. conditional statement two parts: hypothesis and conclusion...

Lesson 2.1 Lesson 2.1 ConditionalConditionalStatementsStatements

Conditional StatementConditional Statement

Two parts: Two parts: hypothesis and hypothesis and conclusionconclusion

If-then formIf-then form

Ex. If it is noon Ex. If it is noon in Philadelphia, in Philadelphia, then it is 9 A.M. then it is 9 A.M. in Los Angeles.in Los Angeles.

““if” contains if” contains hypothesishypothesis

“then” contains “then” contains conclusionconclusion

Example 1Example 1

Write the statements in Write the statements in if-then form.if-then form.

1. A number 1. A number divisible by 9 is divisible by 9 is

also divisible by 3.also divisible by 3.

If a number is If a number is divisible by 9, divisible by 9,

then it is divisible then it is divisible by 3.by 3.

2. All mammals 2. All mammals

breathe oxygenbreathe oxygen..

If an animal is a If an animal is a mammal, then it mammal, then it

breathes oxygen.breathes oxygen.

3. Two points are 3. Two points are collinear if they lie collinear if they lie on the same line.on the same line.

If two points lie If two points lie on the same line, on the same line,

then they are then they are collinear.collinear.

Conditional Conditional Statements can be Statements can be

true or false. If true or false. If they are false, we they are false, we

must find a must find a counterexample.counterexample.

Example 2Example 2

Determine if the statement Determine if the statement is true or false. If it is false, is true or false. If it is false, find a counterexamplefind a counterexample

If a point is If a point is distinct, then it distinct, then it

may lie on more may lie on more than one line.than one line.

If xIf x22=16, then x=4=16, then x=4

If a number is If a number is odd, then it is odd, then it is divisible by 3.divisible by 3.

Converse: formed Converse: formed by switching the by switching the hypothesis and hypothesis and

conclusionconclusion“flip-flop”“flip-flop”

Statement: If you see lightning, then you hear thunder?

Converse: If you hear thunder, then you see lightning.

Are both these true?

If 2 segments are If 2 segments are congruent, then congruent, then they have the they have the same length.same length.

Converse: If two Converse: If two segments have segments have

the same length, the same length, then they are then they are

congruent.congruent.

If an angle is If an angle is acute, then its acute, then its

measure is less measure is less than 90 degrees.than 90 degrees.

Converse: If an Converse: If an angle measures angle measures less than 90less than 90°°, ,

then it is acute.then it is acute.

Inverse: negate Inverse: negate the hypothesis the hypothesis and conclusionand conclusion

Contrapositive: Contrapositive: Negate the Negate the converseconverse

“Negative flip-flop“Negative flip-flop””

If m<A=120If m<A=120°° degrees, then the degrees, then the angle is obtuse.angle is obtuse.

Inverse: Inverse: If m<AIf m<A≠≠120120°°

degrees, then the degrees, then the angle is not obtuse.angle is not obtuse.

Converse:Converse: If the angle is If the angle is obtuse, then obtuse, then m<A=120m<A=120°°..

Contrapositive:Contrapositive: If the angle is not If the angle is not

obtuse, then obtuse, then m<A m<A ≠≠ 120. 120.

Statement: If Statement: If m<P=90m<P=90°°, then , then

<P is a right <P is a right angle.angle.

Inverse:Inverse: If m<P If m<P ≠≠ 90 90°°, ,

then <P is not a then <P is not a right angle.right angle.

Converse:Converse: If <P is a right If <P is a right

angle, then angle, then m<P=90. m<P=90.

Contrapositive:Contrapositive: If <P is not a If <P is not a

right angle, then right angle, then m<P m<P ≠≠ 90. 90.

Statement:Statement:

If an animal is a fish, then it can If an animal is a fish, then it can swim.swim.

InverseInverse

If an animal is not a fish, then it If an animal is not a fish, then it can not swim.can not swim.

ConverseConverse

If an animal can swim, then If an animal can swim, then it is a fish.it is a fish.

ContrapositiveContrapositive

If an animal can’t swim, then If an animal can’t swim, then it is not a fish.it is not a fish.

StatementStatement

If x=y, then 3x=3y.If x=y, then 3x=3y.

InverseInverse

If xIf x≠y, then 3x ≠3y.≠y, then 3x ≠3y.

ConverseConverse

If 3x=3y, then x=y.If 3x=3y, then x=y.

ContrapositiveContrapositive

If 3x If 3x ≠3y, then x ≠y.≠3y, then x ≠y.

Equivalent Equivalent Statements: Statements:

two statements are two statements are both true or both both true or both

false.false.

Postulate 5Postulate 5

Through any 2 points, there Through any 2 points, there exists exactly one line.exists exactly one line.

Postulate 6Postulate 6

A line contains AT LEAST 2 A line contains AT LEAST 2 pointspoints

Postulate 7Postulate 7

If two lines intersect, If two lines intersect, then their intersection then their intersection is exactly one point.is exactly one point.

Postulate 8Postulate 8

Through any 3 Through any 3 NONCOLLINEAR points, NONCOLLINEAR points, there exists exactly one there exists exactly one plane.plane.

Postulate 9Postulate 9

A plane contains AT LEAST A plane contains AT LEAST 3 NONCOLLINEAR points.3 NONCOLLINEAR points.

Postulate 10Postulate 10

If two points lie in a plane, If two points lie in a plane, then the line containing then the line containing them also lies in the plane.them also lies in the plane.

Postulate 11Postulate 11

If two planes intersect, then If two planes intersect, then their intersection is a line.their intersection is a line.

Write postulate 5 in if-then form.Write postulate 5 in if-then form.

Through any 2 points, there exists Through any 2 points, there exists exactly one line.exactly one line.

If there are 2 points, then there If there are 2 points, then there exists exactly one line.exists exactly one line.

InverseInverse

If there is not two points, If there is not two points, then there is not exactly one then there is not exactly one line.line.

ConverseConverse

If there exists exactly one If there exists exactly one line, then there are 2 points.line, then there are 2 points.

ContrapositiveContrapositive

If there is not exactly one If there is not exactly one line, then there is not 2 line, then there is not 2 points.points.

Postulate 8Postulate 8

Through any 3 noncollinear Through any 3 noncollinear points, there exists exactly one points, there exists exactly one plane.plane.

If there are 3 noncollinear points, If there are 3 noncollinear points, then there exists exactly one then there exists exactly one plane.plane.

InverseInverse

If there are not 3 If there are not 3 noncollinear points, then noncollinear points, then there is not exactly one there is not exactly one plane.plane.

ConverseConverse

If there exists exactly one If there exists exactly one plane, then there are 3 plane, then there are 3 noncollinear points.noncollinear points.

ContrapositiveContrapositive

If there is not exactly one If there is not exactly one plane, then there are not 3 plane, then there are not 3 noncollinear points.noncollinear points.