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
