10/21/2015Geometry1 Conditional Statements. 10/21/2015Geometry2 Goals Recognize and analyze a...
Transcript of 10/21/2015Geometry1 Conditional Statements. 10/21/2015Geometry2 Goals Recognize and analyze a...
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Goals
Recognize and analyze a conditional statement
Write postulates about points, lines, and planes using conditional statements
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Conditional Statement
A conditional statement has two parts, a hypothesis and a conclusion.
When conditional statements are written in if-then form, the part after the “if” is the hypothesis, and the part after the “then” is the conclusion.
p → q
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Examples
If you are 13 years old, then you are a teenager.
Hypothesis: You are 13 years old
Conclusion: You are a teenager
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Rewrite in the if-then form
All mammals breathe oxygen If an animal is a mammal, then it
breathes oxygen. A number divisible by 9 is also
divisible by 3 If a number s divisible by 9, then it
is divisible by 3.
CounterexampleCounterexample
Used to show a conditional statement is false.
It must keep the hypothesis true, It must keep the hypothesis true, but the conclusion false!but the conclusion false!
It must keep the hypothesis true, It must keep the hypothesis true, but the conclusion false!but the conclusion false!
It must keep the hypothesis true, It must keep the hypothesis true, but the conclusion false!but the conclusion false!
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Writing a Counterexample
Write a counterexample to show that the following conditional statement is false If x2 = 16, then x = 4. As a counterexample, let x = -4.
The hypothesis is true, but the conclusion is false. Therefore the conditional statement is false.
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Converse
The converse of a conditional is formed by switching the hypothesis and the conclusion.
The converse of p → q is q → p
ConverseConverse
Switch the hypothesis & conclusion parts of a conditional statement.
Ex: Write the converse of “If you are a brunette, then you have brown hair.”
If you have brown hair, then you are a brunette.
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Negation
The negative of the statement Writing the opposite of a statement. Example: Write the negative of the
statement A is acute A is not acute
~p represents “not p” or the negation of p
InverseInverse
Negate the hypothesis & conclusion of a conditional statement.
Ex: Write the inverse of “If you are a brunette, then you have brown hair.”
If you are not a brunette, then you do not have brown hair.
ContrapositiveContrapositive
Negate, then switch the hypothesis & conclusion of a conditional statement.
Ex: Write the contrapositive of “If you are a brunette, then you have brown hair.”
If you do not have brown hair, then you are not a brunette.
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Inverse and Contrapositive Inverse
Negate the hypothesis and the conclusion
The inverse of p → q, is ~p → ~q Contrapositive
Negate the hypothesis and the conclusion of the converse
The contrapositive of p → q, is ~q → ~p.
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Example Write the (a) inverse, (b) converse, and (c)
contrapositive of the statement. If two angles are vertical, then the angles are
congruent. (a) Inverse: If 2 angles are not vertical, then
they are not congruent. (b) Converse: If 2 angles are congruent, then
they are vertical. (c) Contrapositive: If 2 angles are not
congruent, then they are not vertical.
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Equivalent Statements
When 2 statements are both true or both false
A conditional statement is equivalent to its contrapositive.
The inverse and the converse of any conditional are equivalent.
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Point, Line, and Plane Postulates Postulate 5: Through any two points there
exists exactly one line Postulate 6: A line contains at least two
points Postulate 7: If 2 lines intersect, then their
intersection is exactly one point Postulate 8: Through any three noncollinear
points there exists exactly one plane