Corollary

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# Corollary - PowerPoint PPT Presentation

Purpose: Let’s Define Some Terms!!. CONVERSE. Corollary. Proof by Contradiction . If and only if. Axiom . Implies. Converse of a Statement. Statement: If Rex is a dog then Rex is a mammal. (True). Converse: If Rex is a mammal then Rex is a Dog. (False ).

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## PowerPoint Slideshow about 'Corollary' - laqueta

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Presentation Transcript

Purpose: Let’s Define Some Terms!!

CONVERSE

Corollary

If and only if

Axiom

Implies

Converse of a Statement

Statement:IfRex is a dog

thenRex is a mammal. (True)

Converse:IfRex is a mammal

thenRex is a Dog. (False)

What is the Converse of a Theorem?

Consider the sentence below:

If

If

the angles of the shape add up to 180o

the angles of the shape add up to 180o

Then

Then

the shape is a triangle.

the shape is a triangle.

To make the converse statement swap around the

parts of the statement in the green boxes above

This is the converse statement

(TRUE)

Not all converse statements are true.

Consider the sentence below:

If

a shape is a square.

a shape is a square

If

Then

the angles add up to 360o

the angles add up to 360o

Then

Now make the converse statement.

Can you think of a shape with angles of 360o which is not a square ?

Is the Converse True or False?

True

(1) If a triangle has three equal sides then it has three equal angles.

True

(2) If a number is even then the number divides by two exactly.

False

(3) If a shape is a square then the shape has parallel sides.

False

(4) If you have thrown a three and a four then your total score is seven with a die.

Introducing Indirect Proof: Leinster game?

Paul and Mike are driving past the Aviva Stadium. The floodlights are on.

Paul:Are Leinster playing tonight?

Mike:I don’t think so. If a game were being played right now we would see or hear a big crowd but the stands are empty and there isn’t any noise.

Introducing Indirect Proof

Sarah left her house at 9:30 AM and arrived at her aunts house 80 miles away at 10:30 AM.

Use an indirect proof to show that Sarah exceeded the 55 mph speed limit.

If Tim buys two shirts for just over €60, can you prove that at least one of the shirts cost more than €30??

Assume neither shirt costs more than €30

x ≤ 30

+ +

y ≤ 30

x + y ≤ 60

This is a contradiction since we know Tim spent more than

€60

Our original assumption must be false

At least one of the shirts had to have cost more than €30

QED

Triangle ABC has no more than one right angle.

Can you complete a proof by contradiction for this statement?

Assume ∠A and ∠B are right angles

We know ∠A + ∠B + ∠C = 1800

By substitution 900 + 900 + ∠C = 1800

∴ ∠C = 00 which is a contradiction

∴ ∠A and ∠B cannot both be right angles

⇒ A triangle can have at most one right angle

2

1

1

Proof by Contradiction: The Square Root of 2 is Irrational

To prove that 2 is irrational

Assume the contrary: 2 is rational

i.e. there exists integers p and q with no common factors such that:

(Square both sides)

(Multiply both sides by )

(......it’s a multiple of 2)

(......even2 = even)

(Divide both sides by 2)

.