GeometryConditional Statements

 State, use and examine the validity of the converse, inverse, and contrapositive of “if-then” statements.

Conditional Statement

• An if-then statement.

Hypothesis

• The part following if in a conditional statement.

Conclusion

• The part following the then in a conditional statement.

Example Conditional StatementHypothesis, Conclusion

If it rains, then the ground will be wet.

Hypothesis: it rains

Conclusion: the ground will be wet.

Truth value (validity)

• Means every conditional statement is either true or false.

Converse

• Switches the hypothesis and conclusion in of a conditional statement.

Example converse

Conditional Statement:

If it rains, then the ground will be wet.

Converse:

If the ground is wet, then it rained.

1.2Identify 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.

Conclusion: The lines are coplanar.

Hypothesis: Two lines are parallel.

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.

1.2

Write the converse of the conditional: If x = 9, then x + 3 = 12.

The converse of a conditional exchanges the hypothesis and the conclusion.

So the converse is: If x + 3 = 12, then x = 9.

Conditional

Hypothesis Conclusion Hypothesis Conclusion

x = 9 x + 3 = 12 x + 3 = 12 x = 9

Converse

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.

Because 52 = 25, the converse is true.

1.2

The conditional is false/ a counterexample is a = -5; (-5)2 = 25, and –5 5

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.”

1.2

If a circle’s radius is 2 m, then its diameter is 4 m.

1.Identify the hypothesis and conclusion.

2.Write the converse..

3.Determine the truth value of the conditional and its converse.

Show that each conditional is false by finding a counterexample.

4.All numbers containing the digit 0 are divisible by 10.

Hypothesis: A circle’s radius is 2 m. Conclusion: Its diameter is 4 m.

If a circle’s diameter is 4 m, then its radius is 2 m.

Both are true.

Sample: 105

1.2 Questions

Homework

Chapter 2 Section 1 Page 71

1-35, 40-48 54-58

Negation

• Has the opposite truth value.

Inverse

• Negates both the hypothesis and the conclusion.

Contrapositive

• Switches the hypothesis and the conclusion and negates both.

• “the converse of the inverse”

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.”

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.

To write the contrapositive of a conditional, switch the hypothesis and conclusion, then negate both.

Switch and negate both.

Hypothesis Conclusion

Conditional: If ABC is equilateral, then it is isosceles.

Negate both.Inverse: If ABC is not equilateral, then it is not isosceles.

Conditional: If ABC is isosceles, then it is equilateral.

Contrapositive: If ABC is not equilateral, then it is not isosceles.

1.2

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.”

Write the first step of an indirect proof.

Prove: A triangle cannot contain two right angles.

1.2

I and IIP, Q, and R are coplanar and collinear.

Three points that lie on the same line are both coplanar and collinear, so these two statements do not contradict each other.

I and IIP, Q, and R are coplanar and collinear.

Three points that lie on the same line are both coplanar and collinear, so these two statements do not contradict each other.

I and IIIP, Q, and R are coplanar, and m PQR = 60.

Three points that lie on an angle are coplanar, so these two statements do not contradict each other.

Two statements contradict each other when they cannot both be true at the same time.

Examine each pair of statements to see whether they contradict each other.

Identify the two statements that contradict

each other.

I. P, Q, and R are coplanar.II. P, Q, and R are collinear.III. m PQR = 60

I and IIP, Q, and R are coplanar and collinear.

Three points that lie on the same line are both coplanar and collinear, so these two statements do not contradict each other.

I and IIIP, Q, and R are coplanar, and m PQR = 60.

Three points that lie on an angle are coplanar, so these two statements do not contradict each other.

II and IIIP, Q, and R are collinear, and m PQR = 60.

If three distinct points are collinear, they form a straight angle, so m PQR cannot equal 60. Statements II and III contradict each other.

1.2

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.

Write an indirect proof. Prove: ABC cannot contain 2 obtuse angles.

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.

1.2

1. Write the negation of the statement “ D is 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.

For Exercises 3–6, use the following statement:If is parallel to m, then 1 and 2 are supplementary.

3. Write the converse.

4. Write the inverse.

5. Write the contrapositive.

6. Write the first step of an indirect proof.

Assume that 1 and 2 are not supplementary.

If is not parallel to m, then 1 and 2 are not supplementary.

If 1 and 2 are supplementary, then is parallel to m.

If 1 and 2 are not supplementary, then is not parallel to m.

D is not a straight angle.

I and III

1.2 Additional examples

Homework

Chapter 5 Section 4 Page 267

1- 23

Poster Project

Criteria

Pass Objective

Definitions

Examples

1. Converse

2. Inverse

3. Contrapositive

Color