Post on 26-Mar-2015
Using Indirect Reasoning
3 steps to writing an Indirect Proof
What conclusion follows from the pair of statements?
1. Triangle PQR is equilateral
2. Triangle PQR is a right triangle
3. Triangle PQR is isosceles
Identify the pair of statements that form a Contradiction.
1. Triangle PQR is equilateral
2. Triangle PQR is a right triangle
3. Triangle PQR is isosceles
1 & 2
Identify the pair of statements that form a Contradiction.
1. ABCD is a parallelogram.
2. ABCD is a trapezoid.
3. ABCD has two acute angles.
Identify the pair of statements that form a Contradiction.
1. ABCD is a parallelogram.
2. ABCD is a trapezoid.
3. ABCD has two acute angles.
1 & 2
Identify the pair of statements that form a Contradiction.
1. Line l and m are skew.
2. Line l and m do not intersect
3. Line l is parallel to line m.
Identify the pair of statements that form a Contradiction.
1. Line l and m are skew.
2. Line l and m do not intersect
3. Line l is parallel to line m.
1 & 3
Identify the pair of statements that form a Contradiction.
1. Segment FG is parallel to segment KL.
2. Segment FG is perpendicular to segment KL.
3. Segment FG is parallel to segment KL.
Identify the pair of statements that form a Contradiction.
1. Segment FG is parallel to segment KL.
2. Segment FG is perpendicular to segment KL.
3. Segment FG is parallel to segment KL.
1 & 2
Step One
Assume that the opposite of what you want to prove is true.
Step One
Assume that the opposite of what you want to prove is true.
Ex) Statement: It is raining outside
Step One: Indirect Proof
Assume that the opposite of what you want to prove is true.
Ex) Statement: It is raining outside
Assume: It is NOT raining outside.
Examples: Step One
1. <J is not a right angle.
Examples: Step One
1. <J is not a right angle.
Assume <J is a right angle
Examples: Step One
1. Segment YX is congruent to segment AB.
Examples: Step One
1. Segment YX is congruent to segment AB.
Assume Segment YX is not congruent to segment AB.
Examples: Step One
1. Triangle PEN is isosceles.
Examples: Step One
1. Triangle PEN is isosceles.
Assume Triangle PEN is scalene.
Examples: Step One
1. m<2 > 90
Examples: Step One
1. m<2 > 90
Assume m<2 90.
Examples: Step One
1. At least one angle is obtuse
Examples: Step One
1. At least one angle is obtuse
Assume that no angles are obtuse.
Step Two: Indirect Proof
• Use logical reasoning to reach a contradiction of an earlier statement, such as the given information or a theorem. Then state that the assumption you made was false.
Step Two: Indirect Proof
What is the contradiction of step one?
Ex) Statement: It is raining outsideStep One: It is not raining outsideStep Two: The clouds are out and water is coming out of them.
Examples: Step Two
1. What is the contradiction with step one?
Statement: m<2 > 90
Step One: Assume m<2 90.
Step Two: ?
100
Examples: Step Two
1. What is the contradiction with step one?
Statement: m<2 > 90
Step One: Assume m<2 90.
Step Two: The m<2 = 110 which is bigger than 90.
100
Examples: Step Two
What is the contradiction to step one?
2. Triangle PEN is isosceles.
Step One: Assume Triangle PEN is scalene.
P E
N
Examples: Step Two
What is the contradiction to step one?
2. Triangle PEN is isosceles.
Step One: Assume Triangle PEN is scalene.
Step Two: NP and EN are congruent so PEN can’t be scalene.
P E
N
Step 3: Indirect Proof
• State that what you want to prove must be true.
What conclusion follows from the pair of statements?
There are three types of drawbridges: bascule, lift, and swing. This drawbridge does not swing or lift.
What conclusion follows from the pair of statements?
There are three types of drawbridges: bascule, lift, and swing. This drawbridge does not swing or lift.
Conclusion: The bridge is a bascule.
What conclusion follows from the pair of statements?
If this were the day of the party, our friends would be home. No one is home.
What conclusion follows from the pair of statements?
If this were the day of the party, our friends would be home. No one is home.
Conclusion: The party is not today.
What conclusion follows from the pair of statements?
Every air traffic controller in the world speaks English on the job. Sumiko does not speak English.
What conclusion follows from the pair of statements?
Every air traffic controller in the world speaks English on the job. Sumiko does not speak English.
Conclusion: Sumiko is not an air traffic controller.
3 Steps to an Indirect Proof
1. Assume that the opposite of what you want to prove is true.
2. Use logical reasoning to reach a contradiction of an earlier statement, then state that the assumption you made was false.
3. State that what you want to prove must be true.