Transcript Lesson 1 Contents

Lesson 5-3
Indirect Proof
Transparency 5-3
5-Minute Check on Lesson 5-2
Determine the relationship between the lengths of the
given sides.
1. RS, ST
2. RT, ST
Determine the relationship between the measures
of the given angles.
3. A, B
4. B, C
Refer to the figure.
5. Use the Exterior Angle Inequality Theorem to list all
angles whose measures are less than m1.
6.
Standardized Test Practice:
Which angle has the greatest
measure?
A
1
B
2
C
3
D
4
Transparency 5-3
5-Minute Check on Lesson 5-2
Determine the relationship between the lengths of the
given sides.
1. RS, ST
RS < ST
2. RT, ST
RT > ST
Determine the relationship between the measures
of the given angles.
3. A, B
mA < mB
4. B, C
mB < mC
Refer to the figure.
5. Use the Exterior Angle Inequality Theorem to list all
angles whose measures are less than m1.
3, 4, 5, 6
6. Standardized Test Practice: Which angle has the greatest
measure?
A
1
B
2
C
3
D
4
Objectives
• Use indirect proof with algebra
• Use indirect proof with geometry
Vocabulary
• Indirect reasoning – showing something to be
false so that the opposite must be true
• Indirect proof – proving the opposite of what
you assume is true
• Proof by contradiction – proving the
assumption contradicts some fact, definition or
theorem
Key Concept
• Step 1: Assume that the conclusion is false, so then
the opposite is true.
• Step 2: Show that this assumption leads to a
contradiction of the hypothesis, or some other fact,
such as a definition, postulate, theorem or corollary
• Step 3: Point out that because the false conclusion
leads to an incorrect statement, the original
conclusion must be true (the opposite of what we
assumed in step 1)
Algebraic Example
Martha signed up for 3 classes at Wytheville Community
College for a little under $156. There was an
administrative fee of $15, but the class costs varied.
How can you show that at least one class cost less
than $47?
Given: Martha spent less than $156
Prove: At least one class cost (x) less than $47
Step 1: Assume x  $47
Step 2: Then $47 + $47 + $47 + $15  $156
Step 3: This contradicts what Martha paid, so the
assumption must be false.
Therefore one class must cost less than $47!
Geometric Example
K
Given: JKL with side lengths as shown
Prove: mK < mL
8
5
J
7
Step 1: Assume mK  mL
Step 2: By angle-side relationships, JL  JK
Step 3: This contradicts the given side lengths, so
the assumption must be false
Therefore, mK < mL !
L
State the assumption you would make to start
an indirect proof for the statement
is not a
perpendicular bisector.
Answer:
is a perpendicular bisector.
State the assumption you would make to start
an indirect proof for the statement
Answer:
State the assumption you would make to start
an indirect proof for the statement m1 is less
than or equal to m2.
If m1  m2 is false, then m1 > m2.
Answer: m1 > m2
State the assumption you would make to start an
indirect proof for the statement If B is the midpoint of
and
then
is congruent to
The conclusion of the conditional statement is
is
congruent to
The negation of the conclusion is
is
not congruent to
Answer:
is not congruent to
State the assumption you would make to start an
indirect proof of each statement.
a.
is not an altitude.
Answer:
is an altitude.
b.
Answer:
c. mABC is greater than or equal to mXYZ.
Answer: mABC < mXYZ
d. If
is an angle bisector of MLP, then MLH
is congruent to PLH.
Answer: MLH is not congruent to PLH.
Write an indirect proof.
1
Given: ----------- = 20
2y + 4
Prove: y  -2
Indirect Proof:
Step 1 Assume that
.
Step 2 Substitute –2 for y in the equation
Substitution
Multiply.
Add.
This is a contradiction because the
denominator cannot be 0.
Step 3 The assumption leads to a contradiction.
Therefore, the assumption that
must be
false, which means that
must be true.
Write an indirect proof.
Given: ABC with side lengths 8, 10, and 12
as shown.
Prove: mC > mA
Indirect Proof:
Step 1 Assume that
Step 2 By angle-side relationships,
By substitution,
This inequality is a false statement.
Step 3 Since the assumption leads to a contradiction, the
assumption must be false. Therefore, mC > mA.
SHOPPING David bought four new sweaters for a little
under $135. The tax was $7, but the sweater costs
varied. How can you show that at least one of the
sweaters cost less than $32?
Answer:
Given: David spent less than $135.
Prove: At least one of the sweaters x cost less than $32.
That is,
Indirect Proof:
Step 1 Assume that none of the sweaters cost less
than $32.
Step 2
then the minimum total amount David
spent is
However, this is a
contradiction since David spent less than $135.
Step 3 The assumption leads to a contradiction of a
known fact. Therefore, the assumption that
must be false. Thus, at least one of the sweaters
cost less than $32.
Summary & Homework
• Summary:
– In an indirect proof, the conclusion is
assumed to be false and a contradiction is
reached
• Homework:
– pg 258-9: 4-6, 13, 14
Proofs: 11, 22