Section 2-1

Using Deductive Reasoning

If/then statements• Called conditional statements or

simply conditionals.• Have a hypothesis (p) and a

conclusion (q),

"If p, then q:” p q

Example #1• If it is a Friday, then you will

not have homework.

Hypothesis (p): it is a Friday

Conclusion (q):

you will not have homework

Do not include the words “If” and “then” when naming the

hypothesis and conclusion!

Sometimes a conditional

statement is written without using "if"

and "then".

Example #2• Perpendicular lines intersect

at right angles.

Can rewrite:

If two lines are perpendicular, then the lines intersect at right angles.

Hypothesis (p):

Conclusion (q):

two lines are perpendicular

the lines intersect at right angles

If two lines are perpendicular, then the lines intersect at right angles.

converse• formed by interchanging the

hypothesis and conclusion of the conditional.

Conditional : p q

Converse : q p

Example #3• If two lines are

perpendicular, then the lines intersect at right angles.

Write the converse.

If two lines intersect at right angles, then the lines are perpendicular.

•A statement and its converse say different things.

•Some true conditionals may have a false converse.

counterexample• an example that disproves a

statement

–Only need one counterexample to disprove a statement

Example #4• If Angela lives in

Philadelphia, then she lives in Pennsylvania. True

Converse:• If Angela lives in

Pennsylvania, then she lives in Philadelphia. False

Counterexample:She could live in Newtown and still live in PA.

biconditional• A statement that contains

the words “if and only if”p iff q

Iff stands for “if and only if”

Used when a conditional and its converse are both true

Example #5• Congruent segments are

segments that have equal lengths.

Biconditional: Segments are congruent if and only if their lengths are equal.

inverse• the negation of both the

hypothesis and the conclusion of the conditional.–The denial of a statement is called a negation.

Conditional: p q

Inverse: ~ p ~ q Read as not p then not q

Example #6• If two lines are

perpendicular, then the lines intersect at right angles.

If two lines are not perpendicular, then the lines do not intersect at right angles.

Inverse:

contrapositive• negation of both the

hypothesis and conclusion of the converse

Converse: q p

Contrapositive: ~q ~pRead as not q then not p

Example #7• If two lines are

perpendicular, then the lines intersect at right angles.

Write the converse.

If two lines intersect at right angles, then the lines are perpendicular.

Then write the contrapositive.

If two lines do not intersect at right angles, then the lines are not perpendicular.