Reasoning and Conditional Statements

Advanced Geometry

Deductive Reasoning

Lesson 1

Inductive Reasoning

making conclusions based on observations

similar to a hypothesis in science

Conjecture

Examples: Make a conjecture about the next term in each sequence and then find the term.

20, 16, 11, 5, -2, -10160, -80, 40, -20, 10

divide by -2; -5

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.

and are vertical angles.1 2

Counterexamplean 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.Conclusion: The sum is even.

Given: The numbers 2 and 7.Conclusion: The sum is even.

Valid

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: Conclusion: M and N

are vertical angles.

Given: X and Y are vertical anglesConclusion:

M N

X Y

Invalid:

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

(1) If it is Saturday, then I do not have to go to school.

(2) I did not go to school today.

(1) If x = 4, then y = 7.(2) If y = 7, then z = 12.

n is an integer

no valid conclusion

Conditional StatementsExample: If three points are on the same line, then they are collinear.

Example (cont.):

Hypothesis: three points are on the same line

Conclusion: they are collinear

DOES NOT INCLUDE IF

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

then

two angles are adjacent,

they have a common vertex.

Separate the original statement at the verb.

Converse:If two angles have the same measure, then they are congruent.

Original Statement:If two angles are congruent, then they have the same measure.

Converse

switch the hypothesis and conclusion

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.

Example:

All squares are rectangles.