Geometry Ifthen Statements and Postulates Section 2 2

Geometry If-then Statements and Postulates Section 2. 2

If-then Statements and Postulates 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 a baby, then the person is not logical. If a person is not despised, then that person can manage a crocodile. If a person is not logical, then the person is despised. What is the logical conclusion of these statements? 15 -Oct-21 …Geo. Sec 02_02. ppt 2

If-then Statements and Postulates If-then statements are called conditional statements or conditionals. Hypothesis - is the If part (less the if), and the Conclusion - is then part (less 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 15 -Oct-21 …Geo. Sec 02_02. ppt 3

If-then Statements and Postulates Notice that all statements may not be an if-then statement format. 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. 15 -Oct-21 …Geo. Sec 02_02. ppt 4

If-then Statements and Postulates 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. 15 -Oct-21 …Geo. Sec 02_02. ppt 5

If-then Statements and Postulates 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. 15 -Oct-21 …Geo. Sec 02_02. ppt 6

If-then Statements and Postulates 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 15 -Oct-21 …Geo. Sec 02_02. ppt 7

If-then Statements and Postulates Negation - The denial of a statement. The angle is obtuse. The denial of this statement is ; The angle is not obtuse. Other examples of mathematical statements and then their denial Symbolically, p (original), becomes p 15 -Oct-21 …Geo. Sec 02_02. ppt 8

If-then Statements and Postulates The inverse of a statement is formed by negating the hypothesis and the conclusion. Vertical angles are congruent. If two angles are vertical, then they are congruent. and its inverse is, If two angles are NOT vertical, then they are NOT congruent. Symbolically, p q (original), becomes p q 15 -Oct-21 …Geo. Sec 02_02. ppt 9

If-then Statements and Postulates The contrapositive of a statement is formed by negating the hypothesis and the conclusion of the converse. 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. Symbolically, p q (original), becomes q p 15 -Oct-21 …Geo. Sec 02_02. ppt 10

If-then Statements and Postulates 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 15 -Oct-21 …Geo. Sec 02_02. ppt 11

If-then Statements and Postulates If you are 13 years old, then you are a teenager. converse ** If you are a teenager, then you are 13 years old. inverse ** If you are NOT 13 years old, then you are NOT a teenager. contrapositive If you are NOT a teenager, then you are NOT 13 years old. 15 -Oct-21 …Geo. Sec 02_02. ppt 13

If-then Statements and Postulates If an angle measures 90 o, then it is a right angle. converse If an angle is a right angle, then it measures 90 o. inverse If an angle is NOT a right angle, then it does NOT measure 90 o. contrapositive If an angle does NOT measure 90 o, then is NOT a right angle. 15 -Oct-21 …Geo. Sec 02_02. ppt 14

If-then Statements and Postulates 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 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. 15 -Oct-21 …Geo. Sec 02_02. ppt 15

If-then Statements and Postulates END OF LINE 15 -Oct-21 …Geo. Sec 02_02. ppt 16