1. Lesson 2-6 Planning a Proof (page 60)
Essential Question
Can you justify the conclusion of a conditional statement?

2. A proof of a theorem consists of five (5) parts:
Planning a Proof
A proof of a theorem consists of five (5) parts:
Statement of the theorem.
A diagram that illustrates the given information.
A list, in terms of the figure, of what is given.
A list, in terms of the figure, of what you are to prove.
A series of statements and reasons that lead from the given information to the statement that is to be proved.

3. Methods for Writing Proofs:
Gather as much information as you can. Sometimes what you can see will show you a plan. Reread the given. What does it tell you? Look at the diagram. What other information can you conclude?
Work backward . Go to the conclusion, the part you would like to prove. Think … “This conclusion would be true if ? “. “And ? would be true if ? “ and so on, until you have a plan.

4. Suggestions for Writing Proofs:
Follow one of the suggested methods.
Study previous proofs.
Use a pencil.
TRY! TRY! TRY!

5. NOTES
There is usually more than one way to write a proof of a statement.
Be sure that your steps flow logically from one step to the next.
Some steps are more important than others.
Your proof should have enough steps to allow the reader to follow your argument and to see that the statement is indeed true.

6. Theorem 2-7 Supp of ≅ ∠’s R ≅
If two angles are supplements of congruent angles (or of the same angle), then the two angles are congruent.
Given: ∠1 and ∠2 are supplementary;
∠3 and ∠4 are supplementary;
∠2 ≅ ∠4
Prove: ∠1 ≅ ∠3 1 2 3 4

7. Given: ∠1 and ∠2 are supplementary;
∠2 ≅ ∠4
Prove: ∠1 ≅ ∠3 1 2 3 4
Statements Reasons
___________________ ___________________
___________________
______________________ ___________________
See page 61
for the proof
of the first part
of this theorem!

8. Given: ∠1 and ∠2 are supplementary;
∠2 ≅ ∠4
Prove: ∠1 ≅ ∠3 1 2 3 4
Statements Reasons
___________________ ___________________
___________________
______________________ ___________________
∠1 and ∠2 are supplementary; GIVEN
∠3 and ∠4 are supplementary
m∠1 + m∠2 = 180º Def. of Supp. ∠‘s
m∠3 + m∠4 = 180º
m∠1 + m∠2 = m∠3 + m∠4 Substitution Prop.
∠2 ≅ ∠4 , or m∠2 = m∠4 GIVEN
m∠1 = m∠3 , or ∠1 ≅∠3 Subtraction Prop.

9. Given: ∠1 and ∠5 are supplementary; Prove: ∠1 ≅ ∠3
Supp of same ∠ R ≅ 1 3
See page 63
Written Exercises
#15 for second
part of this theorem!
This is part of your
homework.
Statements Reasons
___________________ ___________________
______________________ ___________________

10. Given: ∠2 is supplementary to ∠3 Prove: ∠1 ≅ ∠3
Example:
Given: ∠2 is supplementary to ∠3
Prove: ∠1 ≅ ∠3 1 2 3 4
Statements Reasons
___________________ ___________________
___________________ ___________________
______________________ ___________________
∠2 is supp. to ∠3
Given
m∠1 + m∠2 = 180º
Angle Add. Post.
∠2 is supp. to ∠1
Def. of Supp.∠’s
∠1 ≅ ∠3
Supp. of same ∠R ≅

11. Theorem 2-8 Comp of ≅ ∠’s R ≅ Comp of same ∠ R ≅
If two angles are complements of congruent angles (or of the same angle), then the two angles are congruent.
Given: ∠1 and ∠2 are complementary;
∠3 and ∠4 are complementary;
∠2 ≅ ∠4
Prove: ∠1 ≅ ∠3 1 2 3 4

12. Given: ∠1 and ∠2 are complementary;
∠2 ≅ ∠4
Prove: ∠1 ≅ ∠3 1 2 3 4
Statements Reasons
___________________ ___________________
___________________
______________________ ___________________
∠1 and ∠2 are complementary; GIVEN
∠3 and ∠4 are complementary
m∠1 + m∠2 = 90º Def. of Comp. ∠‘s
m∠3 + m∠4 = 90º
m∠1 + m∠2 = m∠3 + m∠4 Substitution Prop.
∠2 ≅ ∠4 , or m∠2 = m∠4 GIVEN
m∠1 = m∠3 , or ∠1 ≅∠3 Subtraction Prop.

13. Example: Given: LM ⊥ MN and KN ⊥ MN
Name a complement of ∠2: _______
Name a complement of ∠3: _______
What theorem justifies the answers to a & b?
If ∠2 ≅ ∠3, what can be concluded and why? ∠1 ∠4
Ext S 2 adj A∠’s ⊥ ⇒ comp ∠’s
∠1 ≅ ∠4
because comp. of ≅ ∠’s R ≅ M N 2 3 1 4 L K

14. Can you justify the conclusion of a conditional statement?
Assignment
Written Exercises on pages 63 & 64
RECOMMENDED: 15 to 21 odd numbers
REQUIRED: 1 to 13 odd numbers
Quiz on Lesson 2-6
Prepare for a Test on
Chapter 2: Deductive Reasoning
Can you justify the conclusion of a conditional statement?