GEOMETRY 2.1 Conditional Statements
ESSENTIAL QUESTION When is a conditional statement true or false?
WHAT YOU WILL LEARN owrite conditional statements. ouse definitions written as conditional statements. owrite biconditional statements.
CONDITIONAL A type of logical statement that has two parts, a hypothesis and a conclusion. A conditional can be written in IF-THEN form.
SHORTHAND If HYPOTHESIS, then CONCLUSION. If P, then Q. In the study of logic, P s and Q s are universally accepted to represent hypothesis and conclusion.
EXAMPLE 1 If I study hard, then I will get good grades. HYPOTHESIS CONCLUSION
CAN YOU IDENTIFY THE HYPOTHESIS AND CONCLUSION? If today is Monday, then tomorrow is Tuesday. Hypothesis: today is Monday Conclusion: tomorrow is Tuesday. Note: IF is NOT part of the hypothesis, and THEN is NOT part of the conclusion.
YOUR TURN Underline the hypothesis and circle the conclusion. 1. If the weather is warm, then we should go swimming. 2. If you want good service, then take your car to Joe s Service Center.
REWRITING STATEMENTS. ouse common sense. The hypothesis always follows IF. odon t over analyze it. No if? The first part is usually the hypothesis. omake sure the sentence is grammatically correct. Make your English teacher proud! Does it sound right?
EXAMPLE 2A Rewrite the following statement in if-then form: All birds have feathers. What is the hypothesis? What is the conclusion? All birds have feathers If-then form? If an animal is a bird, then it has feathers.
EXAMPLE 2B Rewrite the following statement in if-then form: You are in Texas if you are in Houston. What is the hypothesis? What is the conclusion? You are in Houston You are in Texas If-then form? If you are in Houston, then you are in Texas.
EXAMPLE 2C Rewrite the following statement in if-then form: An even number is divisible by 2. What is the hypothesis? An even number What is the conclusion? Divisible by 2. If-then form? If a number is even, then it is divisible by 2.
YOUR TURN Rewrite the conditional statement in if-then form. 3. Today is Monday if yesterday was Sunday. If yesterday was Sunday, then today is Monday. 4. An object that measures 12 inches is one foot long. If an object measures 12 inches, then it is one foot long.
NEGATION The negative of the original statement. Examples: I am happy. m C = 30. I am not happy. m C 30. A, B and C are on the same line. A, B and C are not on the same line.
NEGATION
EXAMPLE 3 Write the negation of each statement. a. The ball is red. The ball is not red. b. The cat is not black. The cat is black. c. The car is white. The car is not white.
RELATED CONDITIONAL STATEMENTS Looking at the conditional statement: If p, then q. There are three similar statements we can make. o Converse o Inverse o Contrapositive
CONVERSE If Q, then P. The converse of a statement is formed by switching the hypothesis and the conclusion. Conditional: If you play drums, then you are in the band. Converse: If you are in the band, then you play drums.
EXAMPLE 4 Write the converse of the statement below. If you like tennis, then you play on the tennis team. Answer: If you play on the tennis team, then you like tennis.
INVERSE If not P, then not Q. The inverse is formed by taking the negation of the hypothesis and of the conclusion. Conditional: If x = 3, then 2x = 6. Inverse: If x 3, then 2x 6.
EXAMPLE 5 Write the inverse of the statement below. If today is Monday, then tomorrow is Tuesday. Answer: If today is not Monday, then tomorrow is not Tuesday.
CONTRAPOSITIVE If not Q, then not P. The contrapositive is formed by switching and negating the hypothesis and the conclusion. (Take the inverse of the converse, or, the converse of the inverse.) Conditional: If I am in 10 th grade, then I am a sophomore. Contrapositive: If I am not a sophomore, then I am not in 10 th grade.
EXAMPLE 6 Write the contrapositive of the statement below. If x is odd, then x + 1 is even. x is not odd Negate Negate x + 1 is not even If x+1 is not even, then x is not odd.
WARM UP DAY 2 Find the distance between the given points. 1. (3, 8) and ( 2, 7) 2. (5, 4) and ( 1, 3)
LOGICAL STATEMENTS If I live in Mesa, then I live in Arizona. Converse: Inverse: Contrapostive:
YOUR TURN. WRITE THE CONVERSE, INVERSE, AND CONTRAPOSITIVE. If m A = 20, then A is acute. Converse: (switch hypothesis and conclusion) If A is acute, then m A = 20. Inverse: (negate hypothesis and conclusion) If m A 20, then A is not acute. Contrapositive: (switch and negate both) If A is not acute, then m A 20.
REVIEW: LOGICAL STATEMENTS Conditional: If P, then Q. Converse: If Q, then P. Inverse: If not P, then not Q. Contrapositive: If not Q, then not P.
DEFINITION: PERPENDICULAR LINES Two lines that intersect to form a right angle. n Notation: m m n
USING DEFINITIONS You can write a definition as a conditional statement in ifthen form. Let s look at an example: Perpendicular Lines: two lines that intersect to form a right angle. The conditional statement would be: If two lines are perpendicular, then they intersect to form a right angle. The converse statement also ends up being true: If two lines intersect to form a right angle, then they are perpendicular lines.
DAY 2 2.1 Conditional Statements
TRUTH VALUES A conditional is either True or False. To show that it is true, you must have an argument (a proof) that it is true in all cases. To show that it is false, you need to provide at least one counterexample.
EXAMPLE 7 True or false? If false provide a counter example. If x 2 = 9, then x = 3. FALSE! Counterexample: x could be 3.
EXAMPLE 8 If x = 10, then x + 4 = 14. True! Proof: x = 10 x + 4 = 10 + 4 x + 4 = 14
EQUIVALENT STATEMENTS When two statements are both true or both false, they are called equivalent statements. A conditional statement is always equivalent to its contrapositive. The inverse and converse are also equivalent.
EQUIVALENT STATEMENTS Original: If m A = 20, then A is acute. Converse: (switch hypothesis and conclusion) If A is acute, then m A = 20. Inverse: (negate hypothesis and conclusion) If m A 20, then A is not acute. Contrapositive: (switch and negate both) If A is not acute, then m A 20. TRUE False False TRUE
EXAMPLE 9 Statement: If x = 5, then x 2 = 25. TRUE Contrapositive: If x 2 25, then x 5. TRUE Converse: If x 2 = 25, then x = 5. FALSE could be 5. Inverse: If x 5, then x 2 25. FALSE
JUSTIFYING STATEMENTS In math, deciding if a statement is true or false demands that you can justify your answers. Just because, or, It looks like it are not sufficient. Justification must come in the form of Postulates, Definitions, or Theorems. GEOMETRY
EXAMPLE 10 Statement A D, X, and B are collinear. Truth Value D X B TRUE C Reason Definition of collinear points. GEOMETRY
EXAMPLE 11 Statement A AC DB Truth Value D X B TRUE Reason C Definition of Perpendicular lines Def lines GEOMETRY
EXAMPLE 12 Statement D A X B CXB is adjacent to BXA Truth Value TRUE Reason C Def. of adjacent angles Def. of adj. s GEOMETRY
EXAMPLE 13 A Statement DXA and CXB are adjacent angles. Truth Value D X B FALSE Reason C There is not a common side. (Or, they are vertical angles.) GEOMETRY
VERY IMPORTANT! In doing proofs, you must be able to justify every statement with a valid reason. To be able to do this you must know every definition, postulate and theorem. Being able to look them up is no substitute for memorization. GEOMETRY
YOUR TURN A D E B C H F G 3 rd and 6 th hours GEOMETRY
YOUR TURN False (they are not collinear) A D True (sides are opposite rays) E B C H True (post. 8) F G False (no rt. mark) GEOMETRY
YOUR TURN True (def. lines) A D False (they are supplementary) E B C H True (half of 180 is 90 -- a right ) F G GEOMETRY
BICONDITIONALS When a conditional statement and its converse are both TRUE, they can be written as a single biconditional statement. Let s look at an example: Conditional If 2 s are complementary, then their sum is 90. True Converse If the sum of 2 s is 90, then they are complementary. True Biconditional 2 s are complementary if and only if their sum is 90. 4 th 5 th hour
BICONDITIONALS (Continued) Written with p s and q s a biconditional looks like this: p if and only if q. or p iff q. Iff means if and only if.
PUTTING IT ALL TOGETHER Statements In words In symbols Conditional If p, then q p q Converse If q, then p q p Inverse If not p, then not q ~p ~q Contrapostive If not q, then not p ~q ~p Biconditional p if and only if q p q
EXAMPLE 14 Let P be the statement: x = 3 Let Q be the statement: 2x = 6 Write: P Q Q P P Q If x = 3, then 2x = 6. If 2x = 6, then x = 3. x = 3 if and only if 2x = 6. or 2x = 6 iff x = 3. 2.3 DEDUCTIVE REASONING 52
DEFINITIONS ALL definitions are biconditionals. Example: Definition of Congruent Angles Two angles are congruent iff they have the same measure. Conditional: If two angles are congruent, then they have the same measure. Converse: If two angles have the same measure, then they are congruent. GEOMETRY
TRUTH VALUES OF BICONDITIONALS A biconditional is TRUE if both the conditional and the converse are true. A biconditional is FALSE if either the conditional or the converse is false, or both are false. GEOMETRY
EXAMPLE 15 Biconditional x = 5 iff x 2 = 25. Conditional False! True or False? If x = 5, then x 2 = 25. Converse If x 2 = 25, then x = 5. true True or False? False! True or False? GEOMETRY
YOUR TURN Write the following biconditional statement as a conditional statement and its converse. An angle is obtuse iff it measures between 90 and 180. Answer Conditional: If an angle is obtuse, then it measures between 90 and 180. Converse: If an angle measures between 90 and 180, then it is obtuse. GEOMETRY
WHY IS THIS IMPORTANT? Geometry is stated in rules of logic. We use logic to prove things. It teaches us to think clearly and without error. It impresses girl friends (or boy friends). You can talk like