Analyze Conditional Statements Objectives: 1.To write a conditional statement in if-then form 2.To write the negation, converse, inverse, and contrapositive of a conditional statement and identify its truth value 3.To write a biconditional statement

Conditionals Conditionalsif- then Conditionals are statements written in if- then form. A hexagon is a polygon with six sides. SubjectPredicate Ifthen If it is a hexagon, then it is a polygon with six sides. Ifthen If a polygon is a hexagon, then it has six sides. -OR- For clarity: Hypothesis Conclusion

Example 2 Rewrite the conditional statement in if-then form. All 90° angles are right angles.

Example 3 Rewrite the conditional statement in if-then form. Two angles are supplementary if they are a linear pair.

Converse converse The converse of a conditional is formed by reversing the hypothesis (if) and conclusion (then).

Example 4 Write the following statement in if-then form, then write its converse. Is the converse always true? All squares are rectangles.

Truth Value true false A conditional statement can be true or false. TrueTrue: To show that a conditional is true, you have to prove that the conclusion is true every time the hypothesis is satisfied. FalseFalse: To show a conditional is false, you just have to find one example in which the conclusion is not true when the hypothesis is satisfied.

Example 5 What is the opposite of the following statements? 1.The ball is red. 2.The cat is not black.

Negation negation The negation of a statement is the opposite of the original statement. Statement: Statement: The sick boy eats meat. Negation: Negation: The sick boy does not eat meat. Notice that only the verb of the sentence gets negated.

Symbolic Notation Mathematicians are notoriously lazy, creating shorthand symbols for everything. Conditional statements are no different. SymbolConcept p Original Hypothesis q Original Conclusion →“Implies” ~“Not” p → q “ p implies q ” “if p, then q ” ~p~p “not p ”

All Kinds of Conditionals So the symbols make conditionals easy and fun! StatementSymbols Conditional p → q Converse q → p Inverse~ p → ~ q Contrapositive~ q → ~ p

All Kinds of Statements Here are some examples of writing the converse, inverse, and contrapositive of a conditional statement.

Example 6 Write the converse, inverse, and contrapositive of the conditional statement. Indicate the truth value of each statement. If a polygon is regular, then it is equilateral. Which of the statements that you wrote are equivalent?

Equivalent Statements equivalent statements When pairs of statements are both true or both false, they are called equivalent statements. A conditional and its contrapositive are equivalent. An inverse and the converse are equivalent. –So if a conditional is true, so its contrapositive.

Definitions in Geometry reversible In geometry, definitions can be written in if- then form. It is important that these definitions are reversible. In other words, the converse of a definition must also be true. If a polygon is a hexagon, then it has exactly six sides. -AND- If a polygon has exactly six sides, then it is a hexagon.

Perpendicular Lines perpendicular lines If two lines intersect to form a right angle, then they are perpendicular lines.

Example 7 Write the converse of the definition of perpendicular lines. perpendicular lines If two lines intersect to form a right angle, then they are perpendicular lines.

Indirect Proofs When trying to prove a statement is true, it may be beneficial to ask yourself, "What if this statement was not true?" and examine what happens. This is the premise of the Indirect Proof or Proof by Contradiction.

Indirect Proofs Indirect Proof: Assume what you need to prove is false, and then show that something contradictory (absurd) happens.

How to Write an Indirect Proof Step 1: List givens Step 2: Write the negation of what you are trying to prove. Step 3: Use logical reasoning to show that the assumption leads to a contradiction within the proof Step 4: Write what you were originally proving for your final statement, with the reason being contradiction with the appropriate steps that were contradicted. (i.e. contradiction 2,5)

How to Write an Indirect Proof Examples: State the assumption for starting an indirect proof a. b. c. d.

Statement Reason X YZ 1. Given 2. Assumption 3. Sides opposite congruent angles in a triangle are congruent 4. Contradiction 1,3

An Indirect Proof Example You say that my dog, Rex, dug a hole in your yard on July 15 th. I will prove that Rex did not dig a hole in your yard. Let’s temporarily assume that Rex did dig a hole in your yard on July 15 th. Then he would have been in your yard on July 15 th. But this contradicts the fact that Rex was in the kennel from July 14 th to July 17 th. I have bills that show this is true. Thus, our assumption is false, therefore Rex did not dig a hole in your backyard.

Using Indirect Proof Use an indirect proof to prove that a triangle cannot have more than one obtuse angle. SOLUTION: Given ► ∆ABC Prove ►∆ABC does not have more than one obtuse triangle

Using Indirect Proof Step 1: Begin by assuming that ∆ABC does have more than one obtuse angle. –m  A > 90° and m  B > 90° Assume ∆ABC has two obtuse angles. –m  A + m  B > 180° Add the two given inequalities. Step 2: You know; however, that the sum of all the measures of all three angles is 180°. –m  A + m  B +m  C = 180° Triangle Sum Theorem –m  A + m  B = 180° - m  C Subtraction Property of Equality Step 3: So, you can substitute 180° - m  C for m  A + m  B in m  A + m  B > 180° 180° - m  C > 180° Substitution Property of Equality 0° > m  C Simplify