Chapter 2: Reasoning and Proof Section 2.1 - Conditional Statements.

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Presentation transcript:

Chapter 2: Reasoning and Proof Section Conditional Statements

A conditional statement has two parts, a hypothesis and a conclusion. When the statement is written in if-then form, the “if” part is the hypothesis and the “then” part is the conclusion. If it’s noon in GA, then it’s 9 A.M. in CA. Hypothesis Conclusion

Example 1: Rewrite each sentence in “if-then” form. a) Collinear points lie on the same line. If two points are collinear, then they lie on the same line.

b) A number that is divisible by 9 is also divisible by 3 If a number is divisible by 9, then it is divisible by 3. c)All sharks have a boneless skeleton. If a fish is a shark, then it has a boneless skeleton.

*Conditional statements can be either true or false. *To show that a conditional statement is true, show that the conclusion holds for all cases. *To show that a conditional statement is false, show a single counterexample.

Example 2: Write a counterexample to show the following statements are false. a) If a fruit is round, then it is an orange. Counterexample: A lemon b)If x 2 = 16, then x = 4 Counterexample: x can be -4 since (-4) 2 = 16 also.

The converse of a conditional statement is formed by switching the hypothesis and the conclusion. Statement: If you see lightning, then you hear thunder. Converse: If you hear thunder, then you see lightning.

Example 3: Write the converse of the following statement. Statement: If two segments are congruent, then they have the same length. Converse: If two segments have the same length, then the two segments are congruent.

The inverse of a conditional statement is formed by negating the original statement. Statement: If there is a teacher, then the students will learn. Inverse: If there is not a teacher, then the students will not learn.

Example 4: Write the inverse of the statements. a)If there is no sun, then plants will not grow. Inverse: If there is sun, then the plants will grow. b) If the TV is on, then students will not study. Inverse: If the TV is not on, then the students will study.

The contrapositive of a conditional statement is formed by negating the converse of the statement. Statement: If you have the most points, then you win the game. Converse: If you win the game, then you have the most points. Contrapositive: If you do not win the game, then you do not have the most points.

Ex. 5: Write the converse, inverse, and contrapositive of the following statements. 1. Statement: If an angle equals 180°, then the angle is a straight angle. Converse: If an angle is a straight angle, then the angle equals 180°.

Inverse: If an angle is not equal to 180°, then the angle is not a straight angle. Contrapositive: If an angle is not a straight angle, then the angle is not equal to 180°.

2. Statement: If a figure has three sides, then it is a triangle. Converse: If a figure is a triangle, then it has three sides. Inverse: If a figure does not have three sides, then it is not a triangle. Contrapositive: If a figure is not a triangle, then it does not have three sides.

Practice Problems Write the conditional, inverse, converse, and contrapositive statements for each problem. 1)An object weighs one ton if it weighs 2000 pounds. 2)Hagfish live in salt water.

1)Conditional: If an object weighs more than one ton, then it weighs 2000 pounds. Converse: If an object weighs 2000 pounds, then it weighs one ton. Inverse: If an object does not weigh more than one ton, then it does not weigh 2000 pounds. Contrapositive: If an object does not weigh 2000 pounds, then it weighs more than one ton.

2)Conditional: If it is a hagfish, then it lives in salt water. Converse: If it lives in salt water, then it is a hagfish. Inverse: If it is not a hagfish, then it does not live in salt water. Contrapositive: If it does not live in salt water, then it is not a hagfish.

Decide whether the statement is true or false. If false, provide a counterexample. 1)A point may lie in more than one plane. True 2)If x 4 equals 81, then x must equal 3. False (-3) 4 = 81 3)If four points are collinear, then they are coplanar. True

HOMEWORK (Day 1) pg. 75 – 76; 9 – 15, 18 and 19 (write the converse, inverse, and contrapositive of the given statement)

Recall: In Chapter 1, we learned four postulates: Postulate 1: Ruler Postulate Postulate 2: Segment Addition Postulate Postulate 3: Protractor Postulate Postulate 4: Angle Addition Postulate *We are going to learn 7 more postulates and practice writing these postulates in “if-then” form.

Postulate 5 Through any two points there exists exactly one line. If there are two points, then there exists exactly one line through them.

Postulate 6 A line contains at least two points. If a line exists, then it contains as least two points.

Postulate 7 If two lines intersect, then their intersection is exactly one point. Lines m and n are intersecting at point A.

Postulate 8 Through any three noncollinear points there exists exactly one plane. If three points are noncollinear, then there exists exactly one plane through them.

Postulate 9 A plane contains at least three noncollinear points. If a plane exists, then it contains at least three noncollinear points.

Postulate 10 If two points lie in a plane, then the line containing them lies in the plane.

Postulate 11 If two planes intersect, then their intersection is a line.

HOMEWORK (Day 2) pg. 76 ; 25, 27, 29, 32, 35, 36

Worksheet 2.1 Complete #’s 1 – 5 (circle the hypothesis and underline the conclusion), 6, 8, 10, 11, 12, 14, 17, 19 – 22 Staple all work to the worksheet!