2.1 Conditional Statements Geometry 2.1 Conditional Statements

November 10, 2018 2.1 Conditional Statements

Essential Question When is a conditional statement true or false? November 10, 2018 2.1 Conditional Statements

What you will learn Write conditional statements. Use definitions written as conditional statements. Write biconditional statements. November 10, 2018 2.1 Conditional Statements

What is a conditional? November 10, 2018 2.1 Conditional 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. November 10, 2018 2.1 Conditional Statements

Shorthand If P, then Q. If HYPOTHESIS , then CONCLUSION. In the study of logic, P’s and Q’s are universally accepted to represent hypothesis and conclusion. November 10, 2018 2.1 Conditional Statements

Example 1 If I study hard, then I will get good grades. I study hard HYPOTHESIS CONCLUSION November 10, 2018 2.1 Conditional Statements

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. November 10, 2018 2.1 Conditional Statements

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. November 10, 2018 2.1 Conditional Statements

Rewriting statements. Use common sense. Don’t over analyze it. Make sure the sentence is grammatically correct. The hypothesis always follows “IF.” No “if?” The first part is usually the hypothesis. Make your English teacher proud! Does it sound right? November 10, 2018 2.1 Conditional Statements

Example 2a If an animal is a bird, then it has feathers. Rewrite the following statement in if-then form: All birds have feathers. What is the hypothesis? All birds What is the conclusion? have feathers If-then form? If an animal is a bird, then it has feathers. November 10, 2018 2.1 Conditional Statements

Example 2b If you are in Houston, then you are in Texas. Rewrite the following statement in if-then form: You are in Texas if you are in Houston. What is the hypothesis? You are in Houston What is the conclusion? You are in Texas If-then form? If you are in Houston, then you are in Texas. November 10, 2018 2.1 Conditional Statements

Example 2c If a number is even, then it is divisible by 2. 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. November 10, 2018 2.1 Conditional Statements

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. November 10, 2018 2.1 Conditional Statements

Negation The negative of the original statement. Examples: I am happy. I am not happy. mC = 30°. mC  30°. A, B and C are on the same line. A, B and C are not on the same line. November 10, 2018 2.1 Conditional Statements

Negation Think NOT November 10, 2018 2.1 Conditional Statements

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. November 10, 2018 2.1 Conditional Statements

Related Conditionals November 10, 2018 2.1 Conditional Statements

Related conditional statements Looking at the conditional statement: If p, then q. There are three similar statements we can make. Converse Inverse Contrapositive November 10, 2018 2.1 Conditional Statements

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. November 10, 2018 2.1 Conditional Statements

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. November 10, 2018 2.1 Conditional Statements

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. November 10, 2018 2.1 Conditional Statements

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. November 10, 2018 2.1 Conditional Statements

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 10th grade, then I am a sophomore. Contrapositive: If I am not a sophomore, then I am not in 10th grade. November 10, 2018 2.1 Conditional Statements

Example 6 Write the contrapositive of the statement below. If x is odd, then x + 1 is even. Negate Negate x + 1 is not even x is not odd If x+1 is not even, then x is not odd. November 10, 2018 2.1 Conditional Statements

Logical Statements If I live in Mesa, then I live in Arizona. Converse: Inverse: Contrapostive: November 10, 2018 2.1 Conditional Statements

Warm up Day 2 Find the distance between the given points. 1. (𝟑, 𝟖) 𝒂𝒏𝒅 (−𝟐, 𝟕) 2. (𝟓, 𝟒) 𝒂𝒏𝒅 (−𝟏, −𝟑) November 10, 2018 2.1 Conditional Statements

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. November 10, 2018 2.1 Conditional Statements

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. November 10, 2018 2.1 Conditional Statements

Definition: Perpendicular Lines Two lines that intersect to form a right angle. m n Notation: m  n November 10, 2018 2.1 Conditional Statements

Using definitions The conditional statement would be: You can write a definition as a conditional statement in if- then form. Let’s look at an example: The conditional statement would be: The converse statement also ends up being true: Perpendicular Lines: two lines that intersect to form a right angle. If two lines are perpendicular, then they intersect to form a right angle. If two lines intersect to form a right angle, then they are perpendicular lines. November 10, 2018 2.1 Conditional Statements

2.1 Conditional Statements 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. November 10, 2018 2.1 Conditional Statements

Example 7 True or false? If false provide a counter example. If x2= 9, then x = 3. FALSE! Counterexample: x could be –3. November 10, 2018 2.1 Conditional Statements

True! Example 8 If x = 10, then x + 4 = 14. Proof: x = 10 November 10, 2018 2.1 Conditional Statements

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. November 10, 2018 2.1 Conditional Statements

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 November 10, 2018 2.1 Conditional Statements

Example 9 Statement: If x = 5, then x2 = 25. TRUE Contrapositive: If x2  25, then x  5. TRUE Converse: If x2 = 25, then x = 5. FALSE – could be –5. Inverse: If x  5, then x2  25. FALSE November 10, 2018 2.1 Conditional Statements

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. November 10, 2018 Geometry 2.1 Conditional Statements

Example 10 Statement A D, X, and B are collinear. Truth Value D X B TRUE Reason Definition of collinear points. C November 10, 2018 Geometry 2.1 Conditional Statements

Example 11 Def  lines Statement A AC  DB Truth Value D X B TRUE Reason Definition of Perpendicular lines Def  lines C November 10, 2018 Geometry 2.1 Conditional Statements

Example 12 CXB is adjacent to BXA Def. of adjacent angles Statement CXB is adjacent to BXA A Truth Value D X B TRUE Reason Def. of adjacent angles C Def. of adj. s November 10, 2018 Geometry 2.1 Conditional Statements

Example 13 Statement DXA and CXB are adjacent angles. A Truth Value FALSE Reason There is not a common side. (Or, they are vertical angles.) C November 10, 2018 Geometry 2.1 Conditional Statements

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. November 10, 2018 Geometry 2.1 Conditional Statements

Your Turn D A H E B C G F November 10, 2018 3rd hour Your Turn   D A H E B C F G November 10, 2018 Geometry 2.1 Conditional Statements

Your Turn D A False (they are not collinear) H E B C   D A False (they are not collinear) H E B True (sides are opposite rays) C F G True (post. 8) False (no rt.  mark) November 10, 2018 Geometry 2.1 Conditional Statements

Your Turn True (def.  lines) D A False (they are supplementary) H E B   True (def.  lines) D A False (they are supplementary) H E B C True (half of 180 is 90 -- a right ) F G November 10, 2018 Geometry 2.1 Conditional Statements

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°. November 10, 2018 2.1 Conditional Statements

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”. November 10, 2018 2.1 Conditional Statements

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 November 10, 2018 2.1 Conditional Statements

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. November 10, 2018 2.3 Deductive Reasoning

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. November 10, 2018 Geometry 2.1 Conditional Statements

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. November 10, 2018 Geometry 2.1 Conditional Statements

Example 15 False! Biconditional x = 5 iff x2 = 25. Conditional If x = 5, then x2 = 25. Converse If x2 = 25, then x = 5. False! True or False? True or False? true True or False? False! November 10, 2018 Geometry 2.1 Conditional Statements

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. November 10, 2018 Geometry 2.1 Conditional Statements

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… November 10, 2018 2.1 Conditional Statements

November 10, 2018 2.1 Conditional Statements

Assignment 2.1 #2-20 even, 26-38 even, 50, 55, 58 November 10, 2018 2.1 Conditional Statements