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Geometry
Lesson Notes 2.4
Date ________________
Objective: Draw conclusions using Deductive Reasoning. Use the Law of Detachment and
the Law of Syllogism
Deductive reasoning: using facts, rules, definitions, or properties to reach logical
conclusions.
Law of Detachment: given a true conditional statement, and given that the hypothesis is met
by the specific information, then the conclusion is true for the specific information.
If p  q is a true conditional statement and p is satisfied, then q is also true.
Practice:
We know that the product of two negative numbers is positive.
Write a true conditional statement:
If two numbers are negative, then their product is positive.
a. Given: The numbers are −3 and −4.
Conclusion: The product of the −3 and −4 is positive.
Analysis: The conclusion is true.
The hypothesis of the conditional statement is satisfied by the given numbers.
Since the conditional statement is true, and the hypothesis is true, the conclusion
(which follows from the conclusion of the conditional statement) must be true by
the Law of Detachment. Therefore, the product of −3 and −4 must be positive.
b. Given: The product of 2 and 8 is positive.
Conclusion: The numbers 2 and 8 are negative.
Analysis: The conclusion is false.
The conclusion of the conditional statement is satisfied by the given information.
However, having a true conditional statement while the conclusion of the
conditional statement is satisfied does not guarantee that the hypothesis will be
satisfied.
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Example 1 (p 82): Determine Valid Conclusions (with the Law of Detachment)
The following is a true conditional statement. Determine whether each conclusion is
valid based on the given information.
If a point is the midpoint of a segment, then it divides the segment into
two congruent segments.
a. Given: AB  BC
Conclusion: B is the midpoint of AC .
True or False?
b. Given: V is the midpoint of UW .
Conclusion: UV = VW
True or False?
More Practice: Using the Law of Detachment
NOTE: Use the Law of Detachment when you are given a conditional statement and
a fact.
If possible, write a valid conclusion based on the true conditional and the given
information.
a. If an angle measures less than 90, the angle is acute.
Given: mA is 60.
Conclusion: _________________________________________________
b. If a quadrilateral is a square, then it has four congruent sides.
Given: Quadrilateral BCDE has four congruent sides.
Conclusion: _________________________________________________
c. Merrimack High School is locked up at 11pm.
Given: It’s midnight.
Conclusion: _________________________________________________
d. Merrimack High School is locked up at 11pm.
Given: Souhegan High School is locked up.
Conclusion: _________________________________________________
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Law of Syllogism: given two conditional statements such that the conclusion of the first
conditional is the hypothesis of the second conditional, then a new conditional will be
true such that the hypothesis of the first conditional implies the conclusion of the second
conditional.
If pq and qr are true conditional statements, then pr is also true.
Example 2 (p 83): Determine Valid Conclusions From Two Conditionals (Using the Law of
Syllogism)
NOTE: Use the Law of Syllogism when you are given two conditional statements.
Determine if we can make a valid conclusion from the following statements by using the
Law of Syllogism.
a. If a metal is liquid at room temperature, then it is mercury.
If a metal is mercury, then its chemical symbol is Hg.
Identify the p, q and r components of the statements.
Conclusion: __________________________________________________
b. If two angles are adjacent, then they share a common side.
If two angles form a linear pair, then they share a common side.
Conclusion: __________________________________________________
c. The frame shop is closed on holidays.
If the frame shop is closed, no one will be available to answer your call.
Conclusion: __________________________________________________
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E
Example 3 (p 83): Analyzing Conclusions
Determine whether statement (3) follows from statements (1) and (2) by the Law of
Detachment or the Law of Syllogism. If it does, identify the law.
REMEMBER:
Law of Detachment: must have a conditional statement plus a fact
satisfying the hypothesis
Law of Syllogism: must have two conditional statements such that the
conclusion of statement (1) becomes the hypothesis of statement (2)
a. (1) If fossil fuels are burned, then acid rain is produced.
(2) If acid rain falls, wildlife suffers.
(3) If fossil fuels are burned, then wildlife suffers.
__________________________________________________________
b. (1) If x is a real number, then x2 is nonnegative.
(2) x2 is nonnegative.
(3) x is a real number.
__________________________________________________________
c. (1) Football players wear helmets.
(2) Careful bicycle riders wear helmets.
(3) Football players are careful bicycle riders.
__________________________________________________________
d. (1) The measures of the angles in a triangle add up to 180.
(2) ABC is a triangle.
(3) mA + mB + mC = 180
__________________________________________________________
 HW: A5a pp 84-86 (law of detachment) #4-5, 12-19,
(law of syllogism) #6-7, 20-23, (assorted) #8-9, 24-29,
#31, 34, 35
A5b fms-Geometry Worksheet 2.4
A5c 2-4 Skills Practice / Practice
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