2-3 Laws of Logic
The laws of logic can be used to
provide a system for reaching logical
conclusions. This system is called
deductive reasoning. Deductive
reasoning is one of the cornerstones
of the study of geometry.
Law of Detachment
 If p → q is a true conditional
statement and p is true, then q is
true.
p→q
p
q
Example 1
 1. If a student is enrolled at East
Clarendon High, then the student has
an ID number.
 2. Brandon Shaw is enrolled at
East Clarendon High.
 3. Brandon Shaw has an ID
number.
Example 2
 1. If an angle measures 123,
then it is obtuse.
 2. mC = 123.
 3. C is obtuse.
 Determine if statement 3 follows
from statements 1 and 2 by the law
of detachment.
 1. If you are not satisfied with a
tape, then you can return it
within a week for a full refund.
 2. Joe is not satisfied with a tape.
 3. Joe can return the tape within a
week for a full refund.
 1. If a rectangle has four
congruent sides, then it is a
square.
 2. A square has diagonals that
are perpendicular.
 3. A rectangle has diagonals that
are perpendicular.
Law of Syllogism
 If p → q and q → r are true
conditional statements, then p → r is
true.
p→q
q→r
p→r
Example 1
 1. If a number is a whole
number, then it is an integer.
 2. If a number is an integer, then
it is a rational number.
 3. If a number is a whole
number, then it is a rational
number.
Example 2
 1. If fossil fuels are burned, the
acid rain is produced.
 2. If acid rain falls, wildlife
suffers.
 3. If fossil fuels are burned, then
wildlife suffers.
Determine if statement 3
follows from statements 1 and
2 by the law of syllogism.
 1. If Matt lives in Tampa, then he
lives in Florida.
 2. If Matt lives in Florida, then
he lives in the USA.
 3. If Matt lives in Tampa, then he
lives in the USA.
 1. A vocalist can read music.
 2. Ann can read music.
 3. Ann is a vocalist.
Determine if statement (3) follows from
statements (1) and (2) by the Law of Detachment
or the Law of Syllogism.
 If you plan to attend the University of Notre Dame, then
you need to be in the top 10% of your class.
 Rosita Nathan plans to attend Notre Dame.
 Rosita Nathan needs to be in the top 10% of her class.
 If an angle has a measure less than 90, it is acute.
 If an angle is acute, then its supplement is obtuse.
 If an angle has a measure less than 90, then its
supplement is obtuse.
 Vertical angles are congruent.
 <1 = <2
 <1 and <2 are vertical
Determine if a valid conclusion can be reached
from the two true statements using the Law of
Detachment or the law of Syllogism.
 If you want good health, then you
should get 8 hours of sleep each day.
 Patricia Gorman wants good health.
 Patricia Gorman should get 8 hours of
sleep each day.
 Detachment
 If AB = BC and BC = CD,
then AB =CD.
 AB = CD
 No conclusion
 If the measure of an angle is less
than 90, then it is acute.
 If an angle is acute, then it is not
obtuse.
 If the measure of an angle is less
than 90, then it is not obtuse.
 Syllogism
 If there are two points, then there is
exactly one line that contains them.
 There exist two points A and B.
 There is exactly one line that contains
points A and B.
 Detachment.