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GEOMETRY
4-5 Using indirect reasoning
Warm UpComplete each sentence.
1. If the measures of two angles are _____, then the angles
are congruent.
2. If two angles form a ________ , then they are
supplementary.
3. If two angles are complementary to the same angle, then
the two angles are ________ .
equal
linear pair
congruent
Use the given plan to write a two-column proof.
Writing a Two-Column Proof from a Plan
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 m3 + m2 = 180°. By the definition of supplementary angles, 3 and 2 are supplementary.
Writing a Two-Column Proof : Continued
Statements Reasons
1. 1.
2. 2. .
3. . 3.
4. 4.
5. 5.
1 and 2 are supplementary.
1 3
Given
m1 + m2 = 180° Def. of supp. s
m1 = m3
m3 + m2 = 180°
3 and 2 are supplementary
Def. of s
Subst.
Def. of supp. s
Use the given plan to write a two-column proof if one case of Congruent Complements Theorem.
TEACH! Writing a Two-Column Proof
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.
TEACH! Continued
Statements Reasons
1. 1.
2. 2. .
3. . 3.
4. 4.
5. 5.
6. 6.
1 and 2 are complementary.
2 and 3 are complementary.
Given
m1 + m2 = 90° m2 + m3 = 90°
Def. of comp. s
m1 + m2 = m2 + m3
m2 = m2
m1 = m3
Subst.
Reflex. Prop. of =
Subtr. Prop. of =
1 3 Def. of s
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.
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.
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.
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.
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.
Therefore, at least one of the items costs more than $25.
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
Write an indirect proof:
Indirect proof:
Assume has more than one right angle.
Given:
Prove: has at most one right angle.
LMN
LMN
LMN
That is assume are both right angles. and L M
Write an indirect proof:
If are both right angles, then
Given:
Prove: has at most one right angle.
LMN
LMN
According to the Triangle Angle Sum Theorem,.
and L M =m 90om L M
+m 180om L M m N By substitution: 90 +90 180o o om N Solving leaves: 0om N
Write an indirect proof:Given:
Prove: has at most one right angle.
LMN
LMN
If: , This means that there is no triangle LMN. Which contradicts the given statement.
0om N
So the assumption that are both right angles must be false.
and L M
Therefore has at most one right angle.LMN
Lesson Quiz: Part I
Solve each equation. Write a justification for each step.
1.
Lesson Quiz: Part II
Solve each equation. Write a justification for each step.
2. 6r – 3 = –2(r + 1)
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.
Lesson Quiz: Part I
Solve each equation. Write a justification for each step.
1.
z – 5 = –12 Mult. Prop. of =
z = –7 Add. Prop. of =
Given
Lesson Quiz: Part II
Solve each equation. Write a justification for each step.
2. 6r – 3 = –2(r + 1)
Given
6r – 3 = –2r – 2
8r – 3 = –2
Distrib. Prop.
Add. Prop. of =
6r – 3 = –2(r + 1)
8r = 1 Add. Prop. of =
Div. Prop. of =
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.
Trans. Prop. of =
Reflex. Prop. of
Sym. Prop. of
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