Honors Geometry Chapter 2, Section 1 Inductive Reasoning and Conjecturing What do you think? A B C D E F G ___ What would you expect to see next? This educated guess is called a conjecture! A B C D E F G H In this case, we have a counter-example, that shows our original conjecture to be false. You try it!! Given: Points A, B, & C with AB = 15, BC = 12 & AC = 9 Make a conjecture. Some possible conjectures: A, B and C are non-collinear. A, B and C form the vertices of a triangle. One More Time!! Sam was driving home from school and her car died. Make a conjecture about why her car died and then determine a way for Sam to test each conjecture. And yet another! ! Given that points P, Q & R are collinear, Ed made the conjecture that Q is between P & R. Determine if this conjecture is true or false. Explain your answer. Honors Geometry Chapter 2, Section 2 If, then statements and Postulates If, Then Statements If we score more points, then we will win. hypothesis conclusion You try it ! ! Identify the hypothesis and conclusion in the following statements: chickens have lips pigs fly 1) If chickens have lips, then pigs fly. ∠ 2 is a right ∠ m∠2 = 90° 2) If m∠2 = 90°, then ∠2 is a right ∠. I will drive I get my license 3) I will drive, if I get my license. 4) A (Hint: cautious person wears their Try writing in If ... , then ... form first!) seat belt. If a person is cautious, then they wear their seat belt. Conditional Statements If–then statements are also known as conditional statements. This is because we claim that the truth (or falsehood) of the conclusion is conditional upon the truth (or falsehood) of the hypothesis. Some Terminology If A, then B can also be written as A ⇒ B ~A is the opposite of A Example: A is “Dogs are animals.” So: ~A is “Dogs are not animals.” Converse Converse of a Conditional Statement Statement: A ⇒ B Converse: B ⇒ A Statement: If fish swim, then cotton is soft. Converse: If cotton is soft, then fish swim. A⇒B B⇒A Practice ! State the converse of the statement below: If I drive too fast, then I will get a ticket. Converse: If I got a ticket, then I was driving too fast. Inverse Inverse of a Conditional Statement Statement: A ⇒ B Inverse: ~A ⇒ ~B Statement: If fish swim, then cotton is soft. A⇒B Inverse: If fish do not swim, then cotton is not soft. ~A ⇒ ~B Practice ! State the inverse of the statement below: If I drive too fast, then I will get a ticket. Inverse: If I do not drive too fast, then I will not get a ticket. Contrapositive Contrapositive of a Conditional Statement Statement: A ⇒ B Contrapositive: ~B ⇒ ~A Statement: If fish swim, then cotton is soft. A⇒B Converse: If cotton is not soft, then fish do not swim. ~B ⇒ ~A Practice ! State the Contrapositive of the statement below: If I drive too fast, then I will get a ticket. Contrapositive: If I did not get a ticket, then I was not driving too fast. More Postulates Postulate 2–1 Postulate 2–2 Postulate 2–3 Postulate 2–4 Postulate 2–5 Postulate 2–6 Through any two points there is exactly one line. Through any three non-collinear points there is exactly one plane. A line contains at least two points. A plane contains at least three non-collinear points. If 2 pts. lie in a plane, then the entire line containing those pts. . . . lies in the plane. If two planes intersect, then their intersection is a line. Try it !!! The two angles that form a linear pair are supplementary. Write in if–then form If two angles form a linear pair, then they are supplementary. then write the converse If two angles are supplementary, then they form a linear pair. Bi-Conditional Statements A true if - then statement whose converse is also true can be written as a bi-conditional statement. Example If a man is unmarried, then he is a bachelor. If a man is a bachelor, then he is unmarried. becomes A man is unmarried, if and only if, he is a bachelor. or A man is unmarried, iff, he is a bachelor. Bi-Conditional Statements 2 A right angle has a measure of 90°. Write as an if, then statement. If an angle is a right angle, then its measure is 90° Write its converse. If an angle has a measure of 90°, then it is a right angle Write as a bi-conditional statement. An angle is a right angle, iff, its measure is 90° Assignment 2.2 Practice Worksheet

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# Honors Geometry