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Geometry If-then Statements and Postulates Section 2.2

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If-then Statements and Postulates 5-Sep-14…\GeoSec02_02.ppt2 Babies are illogical. Nobody is despised who can manage a crocodile. Illogical persons are despised. Lewis Carroll Rewriting these in an If -then format helps to clarify the preceding statements. If a person is not logical, then the person is despised. If a person is a baby, then the person is not logical. If a person is not despised, then that person can manage a crocodile. What is the logical conclusion of these statements?

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If-then Statements and Postulates 5-Sep-14…\GeoSec02_02.ppt3 If-then statements are called conditional statements or conditionals. Hypothesis - is the If part (less the if), and the Conclusion - is the then part (less the then) of the conditional. If p, then q, where p and q are some statement, is represented symbolically with p q. Symbolically, if p, then q (original), becomes p q

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If-then Statements and Postulates 5-Sep-14…\GeoSec02_02.ppt4 Parallel lines don’t intersect. How would you arrange this to make an if-then statement? If lines are parallel, then they do not intersect. Linear pairs are supplementary. How would you arrange this to make an if-then statement? If two angles are a linear pair, then they are supplementary angles. Notice that all statements may not be an if-then statement format.

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If-then Statements and Postulates 5-Sep-14…\GeoSec02_02.ppt5 When you are converting sentences to if-then statement, look for the key words, if and then. In our examples the if-part is always first, but some sentences may have the if-part at the end of the sentence. For example, I will go to your house, if it rains tomorrow. The hypothesis of this sentence is “it rains tomorrow” and the conclusion is “I will go to your house.” To make this a conditional statement it would be rearranged as follows. If it rains tomorrow, then I will go to your house.

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If-then Statements and Postulates 5-Sep-14…\GeoSec02_02.ppt6 If a sentence has no if-then key words, then use the subject of the sentence as the hypothesis and the object of the sentence as the conclusion. An example is; Babies are illogical. becomes; If a person is a baby, then that person is illogical.

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If-then Statements and Postulates 5-Sep-14…\GeoSec02_02.ppt7 The converse of a statement is when you exchange the hypothesis and the conclusion in a statement. When p q, then the converse is q p. If two lines are perpendicular, then they intersect. and the converse is If two lines intersect, then they are perpendicular. Symbolically, p q (original), becomes q p

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If-then Statements and Postulates 5-Sep-14…\GeoSec02_02.ppt8 Negation - The denial of a statement. The angle is obtuse. The denial of this statement is ; Other examples of mathematical statements and then their denial Symbolically, p (original), becomes p The angle is not obtuse.

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If-then Statements and Postulates 5-Sep-14…\GeoSec02_02.ppt9 The inverse of a statement is formed by negating the hypothesis and the conclusion. Vertical angles are congruent. and its inverse is, If two angles are NOT vertical, then they are NOT congruent. Symbolically, p q (original), becomes p q If two angles are vertical, then they are congruent.

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If-then Statements and Postulates 5-Sep-14…\GeoSec02_02.ppt10 The contrapositive of a statement is formed by negating the hypothesis and the conclusion of the converse. Symbolically, p q (original), becomes q p original If two lines are perpendicular, then they intersect. converse If two lines intersect, then they are perpendicular. contrapositive If two lines do not intersect, then they are not perpendicular.

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If-then Statements and Postulates 5-Sep-14…\GeoSec02_02.ppt11 It can be proven that the contrapositive is logically equivalent to the original statement. A logically equivalent statement has the form of p q q p or if x 2 > 4, then x > 2 if (x > 2), then (x 2 > 4) if x 2, then x 2 4 if (3 = n + 1), then n = 2 if (n=2), then (3 = n + 1) if n 2, then 3 n + 1

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If-then Statements and Postulates 5-Sep-14…\GeoSec02_02.ppt12 Postulate 2-1, Through any two points there is exactly one line. Postulate 2-6, If two planes intersect, then their intersection is a line. Postulate 2-5, If two points lie in a plane, then the entire line containing those two points lies in that plane. Postulate 2-4, A plane contains at least three points not on the same line. Postulate 2-3, A line contains at least two points. Postulate 2-2, Through any three points not on the same line there is exactly one plane.

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If-then Statements and Postulates 5-Sep-14…\GeoSec02_02.ppt13 If you are 13 years old, then you are a teenager. contrapositive If you are NOT a teenager, then you are NOT 13 years old. inverse ** If you are NOT 13 years old, then you are NOT a teenager. converse ** If you are a teenager, then you are 13 years old.

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If-then Statements and Postulates 5-Sep-14…\GeoSec02_02.ppt14 If an angle measures 90 o, then it is a right angle. contrapositive If an angle does NOT measure 90 o, then is NOT a right angle. inverse If an angle is NOT a right angle, then it does NOT measure 90 o. converse If an angle is a right angle, then it measures 90 o.

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If-then Statements and Postulates 5-Sep-14…\GeoSec02_02.ppt15 Summary We converted written statements into conditionals (if-then statements) in order to use logic to determine the validity (truth or falsehood) of those statements. The form of the if-then is if p, then q, or p q, where the if-part of the conditional is the hypothesis and the then-part is the conclusion. We discussed the converse, the negation, the inverse, and the contrapositive of a conditional. We found that the inverse and the converse of a true conditional, is not always itself true. We also discovered that the conditional and its contrapositive are logically equivalent.

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If-then Statements and Postulates 5-Sep-14…\GeoSec02_02.ppt16 END OF LINE

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