Questioning Strategies
Analytic Geometry Student Notes 1.3 & 1.5
AKS: Use inductive reasoning to make conjectures. Use deductive reasoning to
verify conjectures
What should you be able to do?

Use inductive reasoning to identify patterns and make conjectures.

Find counterexamples to disprove conjectures.

Apply the Law of Detachment and the Law of Syllogism in logical reasoning.
1. Inductive reasoning
Identifying patters: Fine the next item in the pattern and explain the reasoning.
1.
January, March, May…
2. 7, 14, 21, 28…
3.
2. Conjecture
Making conjectures: Complete the conjectures.
1.

How many examples do
you need to look at to make
a conjecture? Explain.
3. Counterexample
The sum of two positive numbers is ____________.
2. The number of lines formed by 4 points, no three of which are collinear,
is________.
To show that a conjecture is always true, you must prove it. A counterexample
shows a conjecture is false. A counterexample can be a drawing, a statement,
or a number.
Finding a counterexample: Show that the conjecture is false by finding a
counterexample.
1.

How do you know which
numbers to test when
trying to find a
counterexample for an
algebraic conjecture?
For every integer n, n³ is positive.
2. Two complementary angles are not congruent.
3. For any real number x, x2 ≥ x.
Discussion: Can you prove a conjecture by giving only example in which the conjecture is true? Why or why not?
Questioning Strategies
4. Deductive Reasoning

How can you tell the
difference between inductive
and deductive reasoning?
5. Law of Detachment
Analytic Geometry Student Notes 1.3 & 1.5
Inductive/Deductive Reasoning: Is the conclusion a result of inductive or
deductive reasoning?
a) There is a myth that you can balance an egg on its end only in the spring. A
person was able to balance an egg on July 8, September 21, and December 19.
Therefore this myth is false.
b) There is a myth that the Great Wall of China is the only man-made object visible
from the Moon. The Great Wall is barely visible in photographs taken from 180
miles above Earth. The Moon is about 237,000 miles from Earth. Therefore, the
myth cannot be true.
Law of Detachment: Determine if the conjecture is valid by the Law of Detachment.
a) Given: If the side lengths of a triangle are 5 cm, 12 cm, and 13 cm, then the area
of the triangle is 30 cm2. The area of ∆PQR is 30 cm2.
Conjecture: The side lengths of ∆PQR are 5cm, 12 cm, and 13 cm.
b) Given: In the World Series, if a team wins four games, then the team wins the
series. The Red Sox won four games in the 2004 World Series.
Conjecture: The Red Sox won the 2004 World Series.
6. Law of Syllogism
Law of Syllogism: Determine if the conjecture is valid by the Law of Syllogism.
a) Given: If a figure is a kite, then it is a quadrilateral. If a figure is a
quadrilateral, then it is a polygon.
Conjecture: If a figure is a kite, then it is a polygon.
b) Given: If a number is divisible by 2, then it is even. If a number is even, then it
is an integer.
Conjecture: If a number is an integer, then it is divisible by 2.

When applying the Law of
Syllogism, does the order of
the conditional statements
matter? Explain.
Applying Laws of Deductive Reasoning: Draw a conclusion from the given
information.
a) Given: If 2y = 4, then z = –1. If x + 3 = 12, then 2y = 4. x + 3 = 12
b) If the sum of the measures of two angles is 180°, then the angles are
supplementary. If two angles are supplementary, they are not angles of a
triangle. mA= 135°, and mB= 45°.
How can you recognize when to apply the Law of Detachment verses when to apply the Law of Syllogism?