Warm Up

Determine whether each statement is true or false. If false, give a counterexample.

1. It two angles are complementary, then they are not congruent.

2. If two angles are congruent to the same angle, then they are congruent to each other.

3. Supplementary angles are congruent.

False; only counterexample is 45° and 45°

True

False; one of many counterexamples would be 60° and 120°

Theorem and two-column proof

Vocabulary

When writing a proof, it is important to justify each logical step with a reason. You can use symbols and abbreviations, but they must be clear enough so that anyone who reads your proof will understand them.

Hypothesis Conclusion

• Definitions• Postulates• Properties• Theorems

Write a justif ication for each step,

given that m∠A and m∠B are supplementary and the m∠A=45 0

∠A and ∠B are supplementary

and m∠A=45 0

m∠A +m∠B=180 0

450 +m∠B=180 0

m∠B=135 0

Given information

Def. of Supplementary Angles Substitution Prop of =

(Steps 1, 2)

Subtraction Prop of =

When a justification is based on more than the previous step, you can note this after the reason as in Step 3 of last slide.

Helpful Hint

Write a justif ication for each step, given

that B is the midpoint of AC and AB ≅ EF

B is the midpoint of AC

AB ≅BC

AB ≅EF

BC ≅EF

Given information

Def. of midpoint

Given information

Transitive Property of ≅

A theorem is any statement that you can prove.

Once you have proven a theorem, you can use it as a reason in later proofs.

Key Idea

Each page of your proof notebook includes:

A title (Theorem 2-6-1 Linear Pair Theorem)Write out the theorem itselfIllustrate the theoremProve the theorem – I will prove this one for you!

Linear Pair Theorem (2-6-1)

Def of Straight Angle

Given

Def of Supplementary Angles

Angle Addition Postulate

Def of Straight Angle

If two angles form a linear pair, then they are supplementary. A

D

B C ∠ABD and ∠DBC f orm a linear pair

BAu ruu

and BCu ruu

form a straight angle

m∠ABC =180 0

m∠ABD +m∠DBC =m∠ABC

∠ABD and ∠DBC are supplementary

m∠ABD +m∠DBC =180o Substitution

If two angles are supplementary to the same angle (or to two congruent angles), then the two angles are congruent.

Congruent Supplements Theorem (2-6-2)

Fill in the blanks to complete a two-column proof of one case of the Congruent Supplements Theorem.

∠1 and ∠2 are supplementary and

∠2 and ∠3 are supplementaryGiven:

Prove: ∠1≅∠3

∠1 and ∠2 are supplementary and

∠2 and ∠3 are supplementary

m∠1+m∠2 =m∠2 +m∠3

Subtraction Prop of Equality

∠1≅∠3

1a

3b

5c

6d

Key Idea

Before you start writing a proof, you should plan out your logic. Sometimes you will be given a plan for a more challenging proof. This plan will detail the major steps of the proof for you.

Use the given plan to write a two-column proof.

Plan:

∠1 and ∠2 are supplementary and ∠2 ≅∠3 Given:

Prove: ∠1 and ∠3 are supplementary

Use the def initions of supplementary and congruent angles and

substitution to show that m∠3 +m∠2 =180 0. By the def inition

of supplementary angles, ∠3 and ∠2 are supplementary.

Statements Reasons

1. 1.

2. 2. .

3. . 3.

4. 4.

5. 5.

Given

Def. of supp. angles

Substitution

∠1 and ∠2 are supplementary and ∠2 ≅∠3

m∠1+m∠2 =180 0

m∠2 =m∠3 Def of ≅ ∠'s

m∠1+m∠3 =180 0

∠1 and ∠3 are supplementary Def. of supp. angles

Use the given plan to write a two-column proof of one case of the Congruent Complements Theorem.

∠1 and ∠2 are complementary and

∠2 and ∠3 are complementaryGiven:

Prove: ∠1≅∠3

Plan:

The measures of complementary angles add to 900 by def inition.

Use substitution to show that the sums of both pairs are equal.

Use the subtraction property and the def ition of congruent angles

to conclude that ∠1≅∠3.

Statements Reasons

1. 1.

2. 2. .

3. . 3.

4. 4.

5. 5.

6. 6.

Substitution

Reflex. Prop. of = Subtraction Prop. of =

∠1 and ∠2 are complementary and ∠2 and ∠3 are complementary

m∠1+m∠2 =90 0 and m∠2 +m∠3 =90 0

Def. of comp. angles

Given

m∠1+m∠2 =m∠2 +m∠3

m∠2 =m∠2

m∠1=m∠3

∠1≅∠3 Def of ≅ ∠'s

Write a justif ication for each step, given that the

m∠ABC=90 0and m∠1=4 m∠2( ).

m∠ABC=90 0and m∠1=4 m∠2( )

4 m∠2( ) +m∠2=90 0

m∠1+m∠2=m∠ABC

5 m∠2( )=90

0

m∠2=18 0

Given

Angle Add. Post.

Substitution

Simplify

Division Property of =

Lesson Quiz

Assignment today is page 113: 1-6 and adding theorems 2-6-1 through 2-6-4 to your theorem

notebook.

Remember that homework help is always available at http://www.thinkcentral.com/index.htm

Today’s keyword is “MG7 2-6”.

You need your theorem notebook Monday!