Test corrections Need to write each problem you missed. Including the incorrect answer. Show ALL work!! If you just put the correct letter you will not receive credit. You need to explain why you got the problem wrong. I found the length of the side but did not finish the problem by finding the perimeter I did not know so I guessed. Need to have a parent/guardian sign the test/corrections Due Tuesday Sept. 8th

2.1 Conditional Statements

SWBAT Recognize conditional statements Write converse, negation, inverse, and contrapositive of a conditional statement write biconditionals

If-Then Statements Statements can be changed into If-then statements Football players are jocks. - if you are a football player then you are a jock. Smart kids are geeks. -if you are a smart kid then you are a geek. Boys are mean. - if you are a boy then you are mean. Adjacent angles have a common side. - if angles are adjacent then they have a common side

If-then statements Are used to clarify confusing statements Are called conditional statements or conditionals Hypothesis is the part after the word “If” Conclusion is the part after the word “then” p q represents the conditional statement “if p, then q”

Example 1 Identify the hypothesis and conclusion of the following statements: “if this has been an actual emergency, then the attention signal you just heard would have been followed by official news, information or instruction.” John will advance to the next level of play if he completes the maze on his computer game.

Example 2 Write each statement in “if-then” form and identify the hypothesis and conclusion A five-sided polygon is a pentagon. If a polygon has five sides then it is a pentagon

Counterexample Used to show a conditional statement is false. It must keep the hypothesis true, but the conclusion false!

Ex: Find a counterexample to prove the statement is false. If x2=81, then x must equal 9. counterexample: x could be -9 because (-9)2=81, but x≠9.

Converse Switch the hypothesis & conclusion parts of a conditional statement. q p

Example 3: Write the converse of the true conditional “An angle that measures 120 degrees is obtuse.” Step 1: Write the conditional in if-then form If an angle measures 120 degrees then it is obtuse. Step 2: Write the converse of the true conditional If an angle is obtuse then it measures 120 degrees. Step 3: Determine if your new statement is true or false FALSE Step 4: Give a counterexample if it is not true

Negation The denial of a statement. ~p Ex: negate x=3 x≠3 Ex: negate t>5 t 5

Inverse Negate the hypothesis & conclusion of a conditional statement. The inverse of p q is ~p ~q

Example 4 Write the inverse of the true conditional “vertical angles are congruent.” Determine if the inverse is true or false. If false, give a counterexample. If – then form: If angles are vertical angles then they are congruent Negate both the hypothesis and conclusion If angles are not vertical angles then they are not congruent Determine if they are true or false.

Contrapositive Negating both the hypothesis and conclusion of the CONVERSE The contrapositive of q p is ~q ~p

Example 5 Write the contrapositive of the true conditional “if two angles are vertical then they are congruent.” Statement: Converse: If angles are congruent then they are vertical angles. Contrapositive: If angles are not congruent they are not vertical angles. True or False?

The original conditional statement & its contrapositive will always have the same meaning. The converse & inverse of a conditional statement will always have the same meaning.

Biconditional Joins statements with “if and only if” (iff) Used to write definitions Only true if conditional and the converse are true If both are not true then there is no biconditional

Example 6 Consider the true conditional and the converse, if the converse is also true, combine statements as biconditional. If x = 5, then x + 15 = 20 If x + 15 = 20 then x = 5. X = 5 if and only if x + 15 = 20.

Example 7 Write two statements that form this biconditional A number is divisible by 3 if and only if the sum of its digits is divisible by 3. If a number is divisible by 3, then the sum of its digits is divisible by 3. If the sum of a number’s digits is divisible by 3 then the number is divisible by 3.

Class work Page 71-72 # 2-32 even Page 78 # 3-21 multiples of 3