Download 3.3 Deductive Reasoning Words to know... *Most proofs in geometry

3.3 Deductive Reasoning
Words to know...
*Most proofs in geometry are very much like the proofs in logic.
*They are based on a series of statements that are assumed to be true, called PREMISES or the HYPOTHESIS.
* DEDUCTIVE REASONING: link together the premises and any other true statements to arrive at a true conclusion *The CONCLUSION is sometimes called the DEDUCTION or the deduced statement.
***Since definitions are true statements, they are used in a geometric proof.***
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Proof Form
Given: M is the midpoint of AB
Prove: AM=MB
Statements
Reasons
1.
2.
3.
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Ch 3.4
Direct Proof VS Indirect Proof
‐link together true ‐prove the statement FALSE leading indirectly to the premises and statements conclusion that the negation of that lead directly to a true
the statement must be true
conclusion
Given: ABC is equiangular
Prove: A≅ B≅ C
statements reasons
1. ABC is 1. given
equiangular
2. A≅ B≅ C 2. Definition
of equiangular
triangle.
Given: ABC is equiangular
Prove: A≅ B≅ C
Statements Reasons
1. ABC is 1. Given
equiangular
2. It is NOT the 2. Assumption
case that A≅ B≅ C
3. ABC is NOT 3. If 3 angles
equiangular. of a triangle are NOT congruent then the triangle
is not equiangular
4. A≅ B≅ C 4. Contradiction in 1 and 3 assumption in 2 is false.
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*** Wherever possible, a statement in geometry should be proved by means of a direct proof***
***An indirect proof should be used only when there is no clear way to prove the statement directly***
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Examples:
a) what is the statement
that is to be proved
b) write the assumption
that must appear in an indirect proof
Given: 1 ≅ 2
2 ≅ 3
Prove: 1 ≅ 3
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Examples:
a) what is the statement that is to be proved
b) write the assumption that must appear in an indirect proof
Given: ABC is a scalene triangle
Prove: AB≠BC≠AC
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Examples:
a) what is the statement that is to be proved
b) write the assumption that must appear in an indirect proof
Given: In ABC, CE bisects ACB
Prove: m ACE = m BCE
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Prove Indirectly
Given: DEF is not a straight angle
Prove: ED and EF are not opposite rays
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Homework: pg 109 #1­9
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