Geometry Lombard Gº 12/10/12 HS Page 2
Yash Berwala, Susy Bermudez, Nick Stathos
General form of a conditional statement: If [hypothesis], then [conclusion] --
if A then B.
Converse: If B then A – flip conditional statement so that conclusion becomes the hypotheses and hypothesis becomes the conclusion.
Inverse: if not A, then not B – add “not” to the original conditional statement.
Contrapositive: if not B, then not A (it’s the inverse of the converse – add “not” to the converse)
Example:
Conditional statement / Converse / Inverse / Contrapositive1. If 3x-4 = 11 then x = 5 / If x=5, then 3x-4 = 11 / If 3x-4≠ 11, then x ≠ 5 / If x≠5 then 3x-4≠11
2. If mÐA = 84° then ÐA is not obtuse. / If ÐA is not obtuse, then mÐA = 84°. / If mÐA ≠ 84°, then ÐA is obtuse. / If ÐA is obtuse, then mÐA ≠ 84°.
“Not” in the given conditional statement is cancelled by another “not”: not not obtuse is obtuse. Like when two negatives is a positive.
3. If alternate interior angles are not congruent, then the lines are not parallel. / If lines are not parallel, then alternative interior angles are not congruent. / If alternate interior angles are congruent, then the lines are parallel. / If lines are parallel, then alternate congruent angles are congruent.
Not not parallel is parallel
Indirect Proofs using
v “Indirect” means not direct, not the shortest path from one point to another.
v To prove something indirectly:
Ø Prove the contrapositive. A contrapositive is always true (or false) if the original conditional statement is true (or false).
Ø Contrapositive and conditional statement always have the same “truth value”.
v Steps for indirect proof
1. Assume temporarily that the conclusion – what is to be proved - is false (not true).
2. Reason logically – with facts, theorems, postulates, etc -- until a contradiction of known information is reached.
3. State that the temporary assumption in the step 1 was false, so that the original conclusion must be true by indirect proof.
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