Download Law of Detachment If p q is a true statement and p is true, then q is

Deductive Reasoning
section 2-3
deductive reasoning - the process of using logic to draw conclusions from
given facts, definitions, and properties.
problem # 1 - We hear and read about many myths and legends. Decide whether
each of the following conclusions is based upon inductive or
deductive reasoning:
a. There is a myth that toilets and sinks drain in opposite directions
in the Northern and Southern hemispheres. Someone traveled
to these two locations and noticed that the myth was false.
b. There is a myth that you should not touch a baby bird that has
fallen from its nest because the mother bird will disown it if she
detects human scent. Biologists have shown that birds can not
detect human scent. Therefore, the myth is false.
c. There is a myth that you can balance an egg on its end only on the
spring equinox. A person was able to balance an egg on July 8,
September 21, and December 19. Therefore, this myth is false.
In deductive reasoning, if the given facts are true and you apply the correct
logic, then the conclusion must be true. The Law of Detachment is one valid
form of deductive reasoning.
Law of Detachment
If p
q is a true statement and p is true, then q is true.
problem # 2 - Determine whether each conjecture is valid by the Law of Detachment.
a. If two segments are congruent, then they have the same length.
Given: AB
XY
Conjecture: AB = XY
b. If you are tardy three times, then you must serve detention.
Given: Jenna is in detention.
Conjecture: Jenna was tardy at least three times.
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c. If the length and width of a rectangle are 6 and 3, then its area is 18 u .
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Given: Rectangle ABCD has an area of 18 u .
Conjecture: Rectangle ABCD has a length and a width of 6 and 3.
Another valid form of deductive reasoning is the Law of Syllogism. It allows
you to draw conclusions from two conditional statements when the conclusion
of one is the hypothesis of the other.
Law of Syllogism
If p
q and q
r are true statements, then p
r is a true statement.
problem # 3 - Determine whether each conjecture is valid by the Law of Syllogism.
a. Given: If m A < 90 , then
not a right angle.
Conjecture: If m
A is acute. If
A is acute, then it is
A < 90 , then it is not a right angle.
b. Given: If a number is divisible by 4, then it is divisible by 2.
If a number is even, then it is divisible by 2.
Conjecture: If a number is divisible by 4, then it is even.
c. Given: If an animal is a mammal, then it has hair.
If an animal is a dog, then it is a mammal.
Conjecture: If an animal is a dog, then it has hair.
problem # 4 - If possible, draw a conclusion from the given information in each
case below:
a. If a team wins 10 games, then they play in the finals. If a team
plays in the finals, then they travel to Boston. The Ravens won
10 games.
b. If a team wins 10 games, then they play in the finals. If a team
plays in the finals, then they travel to Boston. The Ravens
traveled to Boston.
c. If a polygon is a rectangle, then it has 4 sides. If a polygon has
4 sides, then it is not a triangle. Polygon P is a rectangle.
d. If the sum of two angles is 180 , then they are supplementary.
If two angles are supplementary, then they can not be angles of
a triangle. m A = 120 , and m B = 60 .