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2.1 & 2.2 notes
Goals1) Describe patterns and use inductive
reasoning.
2) Write definitions as conditional
statements.
Example 1: Describe the pattern
A)
B) -4, 16, -64, -256 …
Inductive Reasoning
• Inductive Reasoning –
• Conjecture –
• Counterexample –
Example 2: Make a conjecture
•
Given the patter of triangles below, make a conjecture about the number
of segments in a similar diagram with 5 triangles (hint: draw pictures if it
is helpful).
Number of
Triangles
Number of
Segments
1
2
3
4
5
Example 3: Find a counterexample
•
Counterexample –
•
Find a counterexample to disprove the conjecture.
A)
Supplementary angles are always adjacent.
B)
If the product of two numbers is positive, then the two numbers
must both be positive.
Conditional Statements
• If-then form
• Hypothesis
• Conclusion
If you are in Ms. Shaw’s class, then you are taking
Geometry.
Example 1: Rewrite a statement in if-then
form
• A) All dogs have four legs.
• B) All 90°angles are right angles.
• C) When n = 8, n2 = 64.
• D) Two angles are congruent if they vertical angles.
Negation
• Negation Statement 1: Her hair is blonde.
Negation 1:
Statement 2: The dog’s name is not Betty.
Negation 2:
Related Conditionals
• Converse:
• Inverse:
• Contrapositive:
Conditional Statement:
Converse-
InverseContrapositive-
If mA  89 , then A is acute.
Example 2: Write four related conditional
statements
• Write the if-then form, the converse, the inverse, and the
contrapositive of the conditional statement. Decide whether each
statement is true or false.
Ms. Shaw’s students are taking a math class.
If-then form:
Converse:
Example 2 continued…
Ms. Shaw’s students are taking a math class.
• Inverse:
• Contrapositive:
Example 3: Write four related conditional
statements
• Write the if-then form, the converse, the inverse, and the
contrapositive of the conditional statement. Decide whether each
statement is true or false.
An equilateral polygon is regular.
If-then form:
Converse:
Example 3 continued…
An equilateral polygon is regular.
• Inverse:
• Contrapositive:
Definitions
• You can write a definition as a conditional statement in __________
or as its ______________.
• Both the conditional statement and its converse are ________.
Perpendicular Lines
Definition:
If two lines intersect to form a right angle,
then they are perpendicular lines.
Example 4: Use definitions
Decide whether each statement about the diagram is true. Explain
your answer using the definitions you have learned.
A)
B)
C)
Biconditional Statements
• Biconditional Statements:
Example:
Rewrite the definition as a biconditonal statement.
Colinear points are points that lie on the same line.
Example 5: Biconditional Statements
• Rewrite the definition as a biconditional statement.
Definition: If the sum of the measures of two angles is 90°, then the
angles are complimentary.