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Warm Up

- Write the converse of each statement. Write each statement as a biconditional where possible.
- If it snows tomorrow, then Ms. Malik will go skiing.
- If you ate too much over Thanksgiving break, then you pants may be tight.
- If two lines are parallel, then they do not intersect.
- If x = - 1, then

- What is the negation of the statement “The 49ers are the best team in the NFL.”

Writing the negation of each statement.

- The m<XYZ is greater than 60.
- Tuesday is not Friday.
- <ABC is obtuse.

Write the Inverse and Contrapositive of the green statement.

Inverse

Contrapositive

Switches the hypothesis and the conclusion and negates both. (converse of inverse)

- Negates both the hypothesis and the conclusion of a conditional statement

- “If you eat a lot of turkey, you will be sleepy.”

- Inverse: If you do not eat a lot of turkey, then you will not be sleepy.
- Contrapositive: If you are not sleepy, then you did not eat a lot of turkey.

Writing an Indirect Proof

Steps:

1. Identify the conjecture to be proven

2. Assume that the opposite is true.

3. Use direct reasoning to show that the assumption leads to a contradiction.

4. Conclude that since the assumption is false, the original conjecture must be true.

Using Indirect Reasoning

Indirect Reasoning: All possibilities are considered and all but one is proved false.

Ex. I yell “Jake, stop disrupting my class!” You look at one Jake, and he’ asleep at his desk, so I must be yelling at the other Jake.

Given: Ms. Malik is a human being.

Prove: Ms. Malik cannot read minds.

Indirect Proof:

Step 1:

- Assume the opposite of our conclusion is true Ms. Malikcan read minds.
Step 2:

- If Ms. Malik can read minds, she should be able to guess any number you can think of.
Can she?

- No! So our assumption that Ms. Malikcan read minds is false.
Step 3: Therefore, Ms. Malik cannot read minds.

Given: ΔJKL Prove: ΔJKL has at most one right angle.

Indirect Proof:

Step 1:

- Assume ΔJKL has more than one right angle.
- So, let’s say <J and <K are both right angles.
Step 2:

- If <J and <K are both right angles, then m<J = 90° &m<K = 90°
- But m<J + m<K + m<L =180° (by the Triangle Angle-Sum Theorem)
- By substituting we get 90° + 90° + m<L = 180°
- Solving the equation leaves m<L = 0°
Can m<L = 0° in ΔJKL?

- No! This contradicts the given statement.
- The assumption that <J and <K are both right angles must be false.
Step 3: Therefore, ΔJKL has at most one right angle.

L

J

Partner Work

- Between you and your partner, choose one of you to be “A”.
- Person A - Think of something that you absolutely know is true.
- Tell it to your Person B.
- Person B – Prove it, using the method we just learned. Assume the opposite is true, then show that it leads to ridiculousness.

Examples

Prove the following using indirect reasoning

(proof by contradiction)

2. An equilateral triangle cannot have a right angle.

Examples

Prove the following using indirect reasoning

(proof by contradiction)

3. A right triangle cannot have an obtuse angle.

Assignment

- P 283 # 1-19 odd, 23

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