Warm Up Name the postulate or theorem that just fies each statement. 1. If a + b = c, then c = a + b 2. If A and B are vertical angles, then A

Warm Up

Name the postulate or theorem that just fies each statement.

1. If a + b = c, then c = a + b

2. If ∠A and ∠B are vertical angles, then ∠A ≅ ∠B.

3. If 2x + 5 = 17, then 2x = 12.

4. If ∠1 and ∠2 form a linear pair, then they are supplementary angles.

Flow Proofs and Paragraph Proofs

Students will learn how to write flow proofs and paragraphs proofs as preparatory activities to writing formal 2-column proofs.

The Point

Not something you're going to be tested onAn aid to writing a formal 2-column proofIf you like writing, the paragraph proof may be

easier for you to start with.If you are a visual person, the flow proof may be

easier to start with.It may turn out that just starting with the 2-column

proof is easier. That's fine. You don't have to use these strategies (except for today).

The Problem

Given: WX = YZProve: WY = XZ

W X Y Z

Paragraph Proof

Given: WX = YZProve: WY = XZ W X Y Z

I know that WX = YZ because it's given.And obviously I know that XY = XY, because everything is equal to itself (that's the reflexive property of equality).So then I can take my WX = YZ and add XY to both sides—that's the addition property of equality.And I end up with WX + XY = YZ + XY.But the Segment Addition Postulate tells me that WX + XY = WY, and also that YZ + XY = XZ.Now I just put those all together, using substitution: WY = XZ. And I'm done!

Flow Proof

Given: WX = YZProve: WY = XZ W X Y Z

GivenWX = YZ WX + XY = YZ + XY

Addition POE

WX + XY = WY

YZ + XY = XZ

WY = XZSubstitution

Seg Addition Post

Seg Addition Post

Another Example

12

Given: ∠1 = 70°

Prove: ∠2 = 110°

Paragraph Proof

12

Given: ∠1 = 70°

Prove: ∠2 = 110°

I know that ∠1 = 70°, because it's given.And I can see from the diagram that ∠1 and ∠2 are a linear pair (so that's given as well).If ∠1 and ∠2 are a linear pair, then they are supplementary. That's the Linear Pair Postulate.And if ∠1 and ∠2 are supplementary, then m∠1 + m∠2 = 180°.But then by the Subtraction POE, m∠2 = 180 – m∠1.So m∠2 = 180 – 70 (that's Substitution POE), and therefore m∠2 = 110°. Done!

Flow Proof

12

Given: m∠1 = 70°

Prove: m∠2 = 110°

m∠1 = 70°

∠1 and ∠2 are an LP

∠1 and ∠2 are supplementary

m∠1 + m∠2 = 180

m∠2 = 180 - m∠1

m∠2 = 180 - 70

m∠2 = 110

You TryGiven: ∠1 and ∠2 are complementary.

∠1 ≅ ∠3, ∠2 ≅ ∠4Prove: ∠3 and ∠4 are complementary.

12

3

4

Paragraph ProofsGiven: ∠1 and ∠2 are complementary.

∠1 ≅ ∠3, ∠2 ≅ ∠4Prove: ∠3 and ∠4 are complementary.

12

3

4

We know that ∠1 and ∠2 are complementary, because it's given.That means that m∠1 + m∠2 = 90, because that's the definition of complementary angles. But it's given that ∠1 ≅ ∠3, so their angle measures are equal by the definition of congruence. Similarly, ∠2 ≅ ∠4, so their measures are equal as well. But then we can substitute m∠3 for m∠1 and m∠2 for m∠4, and we get m∠3 + m∠4 = 90. And therefore, by the definition of complementary angles, ∠3 and ∠4 are complementary.

Flow Proofs

∠1 and ∠2 are complementary

m∠1 + m∠2 = 90

∠1 ≅ ∠3

∠2 ≅ ∠4

m∠1 = m∠3

m∠2 = m∠4

m∠3 + m∠4 = 90

∠3 and ∠4 are complementary

Given Defn of comp ∠'s

Given Defn of Congruence

Substitution Defn of comp ∠'s