GEOMETRY 4-5 Using indirect reasoning Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson...

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Transcript of GEOMETRY 4-5 Using indirect reasoning Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson...

No Slide Title4a. DE = GH, so GH = DE.
4b. 94° = 94°
4d. A Y, so Y A
Sym. Prop. of =
Reflex. Prop. of =
Trans. Prop. of =
Sym. Prop. of
4-5
Complete 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
Using indirect reasoning
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
4-5
Using indirect reasoning
A geometric proof begins with Given and Prove statements, which restate the hypothesis and conclusion of the conjecture.
In a two-column proof, you list the steps of the proof in the left column. You write the matching reason for each step in the right column.
4-5
Given: XY
Reflex. Prop. of =
Using indirect reasoning
If a diagram for a proof is not provided, draw your own and mark the given information on it. But do not mark the information in the Prove statement on it.
Helpful Hint
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.
4-5
1 3
Def. of supp. s
m1 = m3
Def. of s
Using indirect reasoning
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.
4-5
Given
m1 + m2 = 90° m2 + m3 = 90°
Def. of comp. s
m2 = m2
m1 = m3
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.
4-5
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.
4-5
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
4-5
4-5
According to the Triangle Angle Sum Theorem,.
By substitution:
Solving leaves:
Write an indirect proof:
If: , This means that there is no triangle LMN. Which contradicts the given statement.
So the assumption that are both right angles must be false.
4-5
1.
2. 6r – 3 = –2(r + 1)
*
3. x = y and y = z, so x = z.
4. DEF DEF
*
1.
2. 6r – 3 = –2(r + 1)
Given
3. x = y and y = z, so x = z.
4. DEF DEF
Trans. Prop. of =
Reflex. Prop. of
Sym. Prop. of
LMN
LMN
D
D
LMN
D
and
LM
ÐÐ
Given:
LMN
LMN
D
D
and
LM
ÐÐ
=m90
o
mLM
ÐÐ=
+m180
o
mLMmN
LMN
LMN
D
D
LMN
D