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Advanced Geometry
Deductive Reasoning
Lesson 1
Reasoning and
Conditional Statements
Inductive Reasoning
making conclusions based on observations
Conjecture
similar to a hypothesis in science
Examples: Make a conjecture about the next term
in each sequence and then find the term.
160, -80, 40, -20, 10
divide by -2; -5
20, 16, 11, 5, -2, -10
Example: Find the next term in each sequence.
Example: Make a conjecture about the next term
in each sequence and then find the term.
1 1 1 2 5 7
, , , , ,1,
6 3 2 3 6 6
Example: Make a conjecture based on the given
information. Draw a figure to illustrate your
conjecture.
Each side of a square measures 3 feet.
Example: Make a conjecture based on the given
information. Draw a figure to illustrate your
conjecture.
1 and 2 are vertical angles.
Counterexample
an example – proves the statement is false
Example: Give a counterexample to show that the
conjecture is false.
Given: Angles 1 and 2 are adjacent angles.
Conjecture: Angles 1 and 2 form a linear pair.
Example: Determine whether each conjecture is
true or false. Give a counterexample for any
false conjecture.
Given: All sides of a quadrilateral are 3 inches
long.
Conjecture: The quadrilateral’s perimeter is
12 inches.
Deductive Reasoning
making conclusions based on facts
Deductive Reasoning is used to PROVE
statements in mathematics.
All statements must be justified by:
• definitions,
• properties,
• postulates, OR
• theorems
Validity
Definition: being deduced or inferred based
on facts or evidence
Validity and truth are not the same thing.
A statement is valid if it follows the rule.
Example: Determine whether the stated conclusion is
valid based on the given information. If not, write
invalid. Explain your reasoning.
If two numbers are odd, then their sum is even.
Given: The numbers 3 and 11.
Valid
Conclusion: The sum is even.
Given: The numbers 2 and 7.
Conclusion: The sum is even.
Example: Determine whether the stated conclusion is
valid based on the given information. If not, write
invalid. Explain your reasoning.
If two angles are vertical angles, then they are
congruent.
Given: M  N
Conclusion:  M and  N
Invalid:
are vertical angles.
Given:  X and Y are vertical angles
Conclusion: X  Y
Example: Determine a conclusion that follows from
statements (1) and (2). If a valid conclusion does
not follow, write no valid conclusion.
(1) If n is a natural number, then n is an integer.
(2) n is a natural number
n is an integer
(1) If it is Saturday, then I do not have to go to
school.
(2) I did not go to school today.
no valid conclusion
(1) If x = 4, then y = 7.
(2) If y = 7, then z = 12.
Conditional Statements
Example:
If three points are on the same line, then they are collinear.
Example (cont.):
DOES NOT
INCLUDE IF
Hypothesis: three points are on the same line
Conclusion: they are collinear
DOES NOT
INCLUDE THEN
Sometimes a conditional statement is
not written in if-then form.
Example: Write the statement “Adjacent angles have a
common vertex” in if-then form.
If two angles are adjacent,
then they have a common vertex.
Separate the original
statement at the verb.
Converse
switch the hypothesis and conclusion
Original Statement:
If two angles are congruent,
then they have the same measure.
Converse:
If two angles have the same measure,
then they are congruent.
Example:
Write the converse of each conditional. Determine if
the converse is true or false. If it is false, give a
counterexample.
Angles that form a linear pair are supplementary.
All squares are rectangles.