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

Example 1 What are Clairzaps?

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

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

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

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

All squares are rectangles. 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 A conditional statement can be true or false. True: To show that a conditional is true, you have to prove that the conclusion is true every time the hypothesis is satisfied. False: 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? The ball is red. The cat is not black.

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

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

All Kinds of Conditionals So the symbols make conditionals easy and fun! Statement Symbols 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 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 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 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. If two lines intersect to form a right angle, then they are perpendicular lines.

Biconditional A biconditional is a statement that combines a conditional and its true converse in “if and only if” form. 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. A polygon is a hexagon if and only if it has exactly six sides.

Example 8 Write the definition of perpendicular lines as a biconditional statement. If two lines intersect to form a right angle, then they are perpendicular lines.

Exercise 9 Rewrite the definition of right angle as a biconditional statement.

Assignment P. 82-85: 2-10, 19-26, 37, 39, 40, 41 Challenge Problems