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- 1. Geometry Conditional Statements State, use and examine the validity of the converse, inverse, and contrapositive of “if-then” statements.
- 2. Conditional Statement • An if-then statement.
- 3. Hypothesis • The part following if in a conditional statement.
- 4. Conclusion • The part following the then in a conditional statement.
- 5. Example Conditional Statement Hypothesis, Conclusion If it rains, then the ground will be wet. Hypothesis: it rains Conclusion: the ground will be wet.
- 6. Truth value (validity) • Means every conditional statement is either true or false.
- 7. Converse • Switches the hypothesis and conclusion in of a conditional statement.
- 8. Example converse Conditional Statement: If it rains, then the ground will be wet. Converse: If the ground is wet, then it rained.
- 9. 1.2 Identify the hypothesis and the conclusion: If two lines are parallel, then the lines are coplanar. In a conditional statement, the clause after if is the hypothesis and the clause after then is the conclusion. Hypothesis: Two lines are parallel. Conclusion: The lines are coplanar.
- 10. 1.2 Use the Venn diagram below. What does it mean to be inside the large circle but outside the small circle? The large circle contains everyone who lives in Illinois. The small circle contains everyone who lives in Chicago. To be inside the large circle but outside the small circle means that you live in Illinois but outside Chicago.
- 11. 1.2 Conclusion Write the converse of the conditional: x=9 If x = 9, then x + 3 = 12. Converse The converse of a conditional exchanges the hypothesis Hypothesis x + 3 = 12 and the conclusion. Conclusion x + 3 = 12 Conditional So the converse is: If x + 3 = 12, then x = 9. Hypothesis x=9
- 12. 1.2 Write the converse of the conditional, and determine the truth value of each: If a2 = 25, a = 5. Conditional: If a2 = 25, then a = 5. The converse exchanges the hypothesis and conclusion. Converse: If a = 5, then a2 = 25. The conditional is false/ a counterexample is a = -5; (-5)2 = 25, and –5 ≠ 5 Because 52 = 25, the converse is true.
- 13. 1.2 The Mad Hatter states: “You might just as well say that ‘I see what I eat’ is the same thing as ‘I eat what I see’!” Provide a counterexample to show that one of the Mad Hatter’s statements is false. The statement “I eat what I see” written as a conditional statement is “If I see it, then I eat it.” This conditional is false because there are many things you see that you do not eat. One possible counterexample is “I see a car on the road, but I do not eat the car.”
- 14. 1.2 Questions If a circle’s radius is 2 m, then its diameter is 4 m. 1.Identify the hypothesis and conclusion. Hypothesis: A circle’s radius is 2 m. Conclusion: Its diameter is 4 m. 2.Write the converse. If a circle’s diameter.is 4 m, then its radius is 2 m. 3.Determine the truth value of the conditional and its converse. Both are true. Show that each conditional is false by finding a counterexample. 4.All numbers containing the digit 0 are divisible by 10. Sample: 105
- 15. Homework Chapter 2 Section 1 Page 71 1-35, 40-48 54-58
- 16. Negation • Has the opposite truth value.
- 17. Inverse • Negates both the hypothesis and the conclusion.
- 18. Contrapositive • Switches the hypothesis and the conclusion and negates both. • “the converse of the inverse”
- 19. 1.2 Write the negation of “ABCD is not a convex polygon.” The negation of a statement has the opposite truth value. The negation of is not in the original statement removes the word not. The negation of “ABCD is not a convex polygon” is “ABCD is a convex polygon.”
- 20. 1.2 Write the inverse and contrapositive of the conditional statement “If ∆ ABC is equilateral, then it is isosceles.” To write the inverse of a conditional, negate both the hypothesis and the conclusion. Hypothesis Conclusion Conditional: If ABC is equilateral, then it is isosceles. Negate both. Inverse: If ABC is not equilateral, then it is not isosceles. To write the contrapositive of a conditional, switch the hypothesis and conclusion, then negate both. Conditional: If ABC is isosceles, then it is equilateral. Switch and negate both. Contrapositive: If ABC is not equilateral, then it is not isosceles.
- 21. 1.2 Write the first step of an indirect proof. Prove: A triangle cannot contain two right angles. In the first step of an indirect proof, you assume as true the negation of what you want to prove. Because you want to prove that a triangle cannot contain two right angles, you assume that a triangle can contain two right angles. The first step is “Assume that a triangle contains two right angles.”
- 22. 1.2 Identify the two statements that contradict each other. I. P, Q, and R are coplanar. Two statements contradict II. P, Q, and R are collinear. each other when they cannot III. m ∠ PQR = 60 both be true at the same time. Examine each pair of statements to see whether they contradict each other. I Iand II and II I Iand III and III II and III P, Q, and R are P, Q, and R are P, Q, and R are P, Q, and R are P, Q, and R are coplanar and coplanar and coplanar, and coplanar, and collinear, and collinear. collinear. m PQR == 60. m PQR 60. m PQR = 60. Three points that lie Three points that lie Three points that lie Three points that lie If three distinct on the same line are on the same line are on an angle are on an angle are points are collinear, both coplanar and both coplanar and coplanar, so these coplanar, so these they form a straight collinear, so these collinear, so these two statements do two statements do angle, so m PQR two statements do two statements do not contradict each not contradict each cannot equal 60. not contradict each not contradict each other. other. Statements II and III other. other. contradict each other.
- 23. 1.2 Write an indirect proof. Prove: ∆ABC cannot contain 2 obtuse angles. Step 1: Assume as true the opposite of what you want to prove. That is, assume that ∆ ABC contains two obtuse angles. Let ∠A and ∠B be obtuse. Step 2: If ∠ A and ∠ B are obtuse, m ∠ A > 90 and m ∠ B > 90, so m ∠A + m ∠B > 180. Because m ∠ C > 0, this means that m ∠ A + m ∠ B + m ∠ C > 180. This contradicts the Triangle Angle-Sum Theorem, which states that m ∠ A + m ∠ B + m ∠ C = 180. Step 3: The assumption in Step 1 must be false. ∆ ABC cannot contain 2 obtuse angles.
- 24. 1.2 Additional examples 1. Write the negation of the statement “ D is a straight angle.” D is not a straight angle. 2. Identify two statements that contradict each other. I. x and y are perfect squares. II. x and y are odd. III. x and y are prime. I and III For Exercises 3–6, use the following statement: If is parallel to m, then 1 and 2 are supplementary. 3. Write the converse. If 1 and 2 are supplementary, then is parallel to m. 4. Write the inverse. If is not parallel to m, then 1 and 2 are not supplementary. 5. Write the contrapositive. If 1 and 2 are not supplementary, then is not parallel to m. 6. Write the first step of an indirect proof. Assume that 1 and 2 are not supplementary.
- 25. Homework Chapter 5 Section 4 Page 267 1- 23
- 26. Poster Project Criteria Pass Objective Definitions Examples 1. Converse 2. Inverse 3. Contrapositive Color