2-4 Deductive Reasoning
Ms. Andrejko
Real World
Vocabulary
 Deductive reasoning- uses facts, rules, definitions, or
properties to reach logical conclusions from given statements
 Unlike inductive reasoning which uses patterns/observations
 Law of Detachment- if p  q is a true conditional, and p
is true, then q is is also true
 Law of Syllogism- if p  q and q  r are true statements,
then p  r is a true statement
 Note: Stating a counterexample is not a valid method of
proving a conjecture.
Laws
 Detachment:
 If a car is out of gas, then it will not start. Sarah’s car
is out of gas.
 Valid conclusion: Sarah’s car will not start.
 Syllogism:
 If you get a job, then you will earn money. If you earn
money, then you will buy a car.
 Valid conclusion: If you get a job, then you will buy a car.
Examples – Inductive or deductive?
 1. A dental assistant notices a patient has never been on time
for an appointment, he concludes the patient will be late for
her next appointment.
Inductive
 2. If Eduardo decides to go to a concert tonight, he will miss
the football game. Tonight Eduardo went to a concert.
Eduardo missed football practice.
Deductive
Practice– Inductive or deductive?
 1. A person must be insured in order to drive a car. Andy is
driving a car. Andy is insured.
Deductive
 2. Whenever Julie attends a tutoring session she notices that
her grades have improved. Julia attends a tutoring session
and she concludes that her grades will improve.
Inductive
Examples – Valid or Invalid?
Given: If the sum of the measures of two angles is 180, then the
angles are supplementary.
VALID – Law of detachment
m∠A + m∠B is 180.
Conclusion: ∠A and ∠B are supplementary.
1.
2. Given: If the sum of the measures of two angles is 90, then the
angles are complementary.
m∠ABC is 45 and m∠DEF is 48.
Conclusion: ∠ABC and ∠DEF are complementary.
INVALID: 45 + 48 = 93 , 93 ≠ 90.
Practice– Valid or Invalid?
Given: If a point is the midpoint of a segment, then it divides the
segment into two congruent segments.
R is the midpoint of QS
VALID – Law of detachment
Conclusion: QR  QS
1.

2.
Given: If 3 points are noncollinear, they determine a plane.

Points A,B, and C, lie in plane G.
Conclusion: Points A, B, and C are noncollinear.
INVALID: points A, B, and C, could be
collinear and form a line.
Examples- Law of Syllogism
1.
If two angles are complementary, then the sum of their
measures is 90.If the sum of the measures of two angles is
90, then both of the angles are acute.
If 2 angles are complementary, then both
of the angles are acute.
2.
If the heat wave continues, then air conditioning will be
used more frequently. If air conditioning is used more
frequently, then energy costs will be higher.
If the heat wave continues, then energy
costs will be higher.
Examples- Law of Syllogism
 3. If it is Tuesday, then Marla tutors chemistry. If Marla tutors
chemistry, then she arrives home at 4 P.M.
If it’s a Tuesday, then Marla arrives home at 4.
 4. If two lines are perpendicular, then they intersect to form
right angles. Lines r and s form right angles.
NO VALID CONCLUSION
 5. If two angles form a linear pair, then the two angles are
supplementary. If two angles are supplementary, then the
sum of their measures is 180.
If 2 angles form a linear pair, then the sum of
their measures is 180.
FOLDABLE
LOGIC
TAB