Download Geometry_CH-04_Lesson-5 _Using Indirect Reasoning _ Geometric

4-5
Using indirect reasoning
Warm Up
Lesson Presentation
Lesson Quiz
GEOMETRY
4-5
Using indirect reasoning
Warm Up
Complete each sentence.
equal
1. If the measures of two angles are _____, then the angles
are congruent.
2. If two angles form a
linear pair
________ , then they are
supplementary.
3. If two angles are complementary to the same angle, then
congruent
the two angles are ________ .
GEOMETRY
4-5
Using indirect reasoning
Writing a Two-Column Proof from a Plan
Use the given plan to write a two-column proof.
Given: 1 and 2 are supplementary, and
1  3
Prove: 3 and 2 are supplementary.
Plan: Use the definitions of supplementary and congruent
angles and substitution to show that m3 + m2 = 180°.
By the definition of supplementary angles, 3 and 2 are
supplementary.
GEOMETRY
4-5
Using indirect reasoning
Writing a Two-Column Proof : Continued
Statements
Reasons
1. 1 and 2 are supplementary.1. Given
1  3
2. m1 + m2 = 180°
2. .Def. of supp. s
= m3
3. m1
.
3. Def. of  s
4. m3 + m2 = 180°
4. Subst.
5. 3 and 2 are supplementary5. Def. of supp. s
GEOMETRY
4-5
Using indirect reasoning
TEACH! Writing a Two-Column Proof
Use the given plan to write a two-column proof if one
case of Congruent Complements Theorem.
Given: 1 and 2 are complementary, and
2 and 3 are complementary.
Prove: 1  3
Plan: The measures of complementary angles add to 90° by
definition. Use substitution to show that the sums of both
pairs are equal. Use the Subtraction Property and the
definition of congruent angles to conclude that 1  3.
GEOMETRY
4-5
Using indirect reasoning
TEACH! Continued
Statements
Reasons
1. 1 and 2 are complementary. 1. Given
2 and 3 are complementary.
2. m1 + m2 = 90°
m2 + m3 = 90°
of comp. s
2. Def.
.
+ m2 = m2 + m3
3. m1
.
3. Subst.
4. m2 = m2
4. Reflex. Prop. of =
5. m1 = m3
5. Subtr. Prop. of =
6. 1  3
6. Def. of  s
GEOMETRY
4-5
Using indirect reasoning
Use indirect reasoning to prove:
If Jacky spends more than $50 to buy two items at a
bicycle shop, then at least one of the items costs
more than $25.
Given: the cost of two items is more than $50.
Prove: At least one of the items costs more than $25.
Begin by assuming that the opposite is true. That is
assume that neither item costs more than $25.
GEOMETRY
4-5
Using indirect reasoning
Use indirect reasoning to prove:
If Jacky spends more than $50 to buy two items at a
bicycle shop, then at least one of the items costs
more than $25.
Given: the cost of two items is more than $50.
Prove: At least one of the items costs more than $25.
Begin by assuming that the opposite is true. That is
This
means
bothitem
items
costmore
$25 or
less.
This
assume
thatthat
neither
costs
than
$25.
means that the two items together cost $50 or less.
This contradicts the given information that the
amount spent is more than $50. So the assumption
that neither items cost more than $25 must be
incorrect.
GEOMETRY
4-5
Using indirect reasoning
Use indirect reasoning to prove:
If Jacky spends more than $50 to buy two items at a
bicycle shop, then at least one of the items costs
more than $25.
Therefore, at least one of the items costs more
than $25.
This means that both items cost $25 or less. This
means that the two items together cost $50 or less.
This contradicts the given information that the
amount spent is more than $50. So the assumption
that neither items cost more than $25 must be
incorrect.
GEOMETRY
4-5
Using indirect reasoning
Writing an indirect proof
Step-1: Assume that the opposite of
what you want to prove is true.
Step-2: Use logical reasoning to reach a
contradiction to the earlier statement,
such as the given information or a
theorem. Then state that the
assumption you made was false.
Step-3: State that what you wanted to
prove must be true
GEOMETRY
4-5
Using indirect reasoning
Write an indirect proof:
Given: LMN
Prove: LMN has at most one right angle.
Indirect proof:
Assume LMN has more than one right angle.
That is assume L
and M
are both right angles.
GEOMETRY
4-5
Using indirect reasoning
Write an indirect proof:
Given: LMN
Prove: LMN has at most one right angle.
If
L and M are both right angles, then
o
mL =mM  90
According to the Triangle Angle Sum Theorem,.
mL +mM  mN  180
o
o
o
By substitution:
90 +90  mN  180
o
Solving leaves: mN  0
o
GEOMETRY
4-5
Using indirect reasoning
Write an indirect proof:
Given: LMN
Prove: LMN has at most one right angle.
If: mN  0 , This means that there is no
triangle LMN. Which contradicts the given statement.
o
So the assumption that L
right angles must be false.
and M
are both
Therefore LMN has at most one right angle.
GEOMETRY
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Using indirect reasoning
GEOMETRY
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Using indirect reasoning
Lesson Quiz: Part I
Solve each equation. Write a justification for
each step.
1.
GEOMETRY
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Using indirect reasoning
Lesson Quiz: Part II
Solve each equation. Write a justification for
each step.
2. 6r – 3 = –2(r + 1)
GEOMETRY
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Using indirect reasoning
Lesson Quiz: Part III
Identify the property that justifies each
statement.
3. x = y and y = z, so x = z.
4. DEF  DEF
5. AB  CD, so CD  AB.
GEOMETRY
4-5
Using indirect reasoning
Lesson Quiz: Part I
Solve each equation. Write a justification for
each step.
1.
Given
z – 5 = –12
z = –7
Mult. Prop. of =
Add. Prop. of =
GEOMETRY
4-5
Using indirect reasoning
Lesson Quiz: Part II
Solve each equation. Write a justification for
each step.
2. 6r – 3 = –2(r + 1)
6r – 3 = –2(r + 1) Given
6r – 3 = –2r – 2
Distrib. Prop.
8r – 3 = –2
8r = 1
Add. Prop. of =
Add. Prop. of =
Div. Prop. of =
GEOMETRY
4-5
Using indirect reasoning
Lesson Quiz: Part III
Identify the property that justifies each
statement.
3. x = y and y = z, so x = z. Trans. Prop. of =
4. DEF  DEF
Reflex. Prop. of 
5. AB  CD, so CD  AB.
Sym. Prop. of 
GEOMETRY