When you make a conclusion based on a pattern of examples or past events, you are using
Inductive Reasoning
A conclusion that you reach based on inductive reasoning
Conjecture
An educated guess
Conjecture
Explain why a conjecture can be true or false
It is an educated guess, and just because you say yes or no does not mean that the guess is correct.
If you find ____ that does not follow the conjecture, then the conjecture is false
One example
If you find one example that does not follow the conjecture, then the conjecture is _____.
False
What is the false example called for a conjecture?
Counterexample
What joins two statements based on a condition?
If-then statements
If-then statements join two statements based on what?
A condition
If-then statements are also called...
Conditional Statements
In a conditional statement, the part following if is the
Hypothesis
In a conditional statement, the part following then is the
Conclusion
Are there different ways to express a conditional statement?
YES
Formed by exchanging the hypothesis and the conclusion
Converse
What is the converse?
The conditional statement flipped...
Conclusion first, hypothesis next
Write the converse of this statement:
-If a figure is a triangle, then it has three angles
Converse:
If the figure has three angles, then it is a triangle
Has opposite truth value of a given statement.
Negation
Negates both hypothesis and conclusion of a conditional
Inverse
The inverse of the converse of a conditional
Contrapositive
Statements with the same truth value
Equivalent Statements
What is the symbolic form of conditional?
p yields q
What is the way you read a conditional?
If p, then q
What is the symbolic form of inverse?
~p yields ~q
What is the way you read an inverse?
If not p, then not q
What is the symbol for negation?
~p
What is the way you read negation?
Not p
What is the symbolic form of contrapositive?
~q yields ~p
What is the way you read contrapositive?
If not q, then not p
If the conditional is true, what else would also be true?
The contrapositive
What would have to be true in order for the contrapositive to be true?
Conditional statement
If the conditional is false, what else would be false?
The contrapositive
If statements have the same truth value, then they are known as
Equivalent Statements
When are two statements equivalent statements?
When they have the same truth value
When a conditional and its converse are true, what can you combine them as?
True Biconditionals
When a conditional and its converse are true and you combine them, what phrase do you use?
'If and only if'
How do you write a biconditional?
-Take the conditional
-Write the converse of the conditional
-if converse is true, write the conditional statement using if and only if to become conditional
Write the biconditional of:
If x=5, then x+15=20
Converse: True ?
If x+15=20, then x=5.
Biconditional:
x=5 if and only if x+15=20
What parts does a biconditional separate into?
-The conditional
-The converse
Separate this biconditional into parts:
Lines are skew if and only if they are non coplanar.
-If lines are skew, then they are non coplanar.
-If lines are noncoplanar, then they are skew.
What is a good definition
A statement that can HELP identify or classify an object
A statement that can help identify or classify an object.
Good Definition
To be a good definition, you need (3)...
-Use commonly understood terms
-Be precise and not vague
-Be reversible, can write as a true biconditional
How do you show that the definition is reversible?
Write:
-Conditional
-Converse
-Biconditional
Process of reasoning logically from given statement to form a conclusion
Deductive Reasoning
What is deductive reasoning also called?
Logical reasoning
If a conditional is true, and its hypothesis is true, then what else is true?
The conclusion
If p yields q is a true statement and p is true, then what else is true in this?
q, in the Law of Detachment
If a conditional is true, and the hypothesis is true, then the conclusion is also true
Law of Detachment
If p yields q is a true statement and p is true, then q is true...
Law of Detachment
If p yields q and q yields r are true statements, then p yields r is a true statement...
Law of Syllogism
If p yields q and q yields r are true statements, then what else would be true?
p yields r.
What is one way to think about the Law of Syllogism?
You eliminate the sylla (simi) similar parts that the statements have in common and are left with your statement
When you interpret a diagram, you can only assume information about size or measure if...
It is marked
What is the reflexive property?
a=a
Easy way to remember reflexive property?
It is reflecting itself, reflexive off a mirror
What is the symmetric property?
If a=b, then b=a
What property:
a=a?
Reflexive Property of Equality
What property:
If a=b, then b=a?
Symmetric Property of Equality
What is the transitive property?
If a=b and b=c, then a=c.
What property:
If a=b and b=c, then a=c?
Transitive Property of Equality
What is the substitution property?
If a=b, then b can replace a in any expression
What property:
If a=b, then b can replace a in any expression
Substitution Property of Equality
Logical argument that shows a statement is true
Proof
Has numbered statements and corresponding reasons that show an argument in logical order
Two-Column Proof
A statement that can be proven
Theorem
Segment, angle congruence is what?
-Reflexive
-Symmetric
-Transitive
This is reflexive, symmetric, and transitive..
-Segment Congruence
-Angle Congruence
Congruent segments have the same___
Measure
Besides congruent segments having the same measure, what else has the same measure?
Congruent angles
Two angles are congruent if and only if they have the same..
Degree Measure
To show that angles are congruent to each other, these are used..
Arcs
To show that there is a second set of congruent angles, what is used?
Double arcs
All right angles are ____.
Congruent
When two lines intersect, _____ angles are formed.
Four
Two pair of nonadjacent angles:
Vertical angles
Two nonadjacent angles formed by a pair of intersecting lines
Vertical angles
What makes angles vertical?
If there are two nonadjacent angles formed by a pair of intersecting lines
Adjacent means..
Next to
Angles that:
-Share a common side
-Have the same vertex
-Have no interior points in common
Adjacent angles
What are adjacent angles?
Angles that:
-Share a common side
-Have the same vertex
-Have no interior points in common
The sum of their degree measure is 90.
Complementary Angles
What makes complementary angles?
Two angles whose sum of their degree measure is 90.
If the sum of the measure of two angles in 180, they are
Supplementary Angles
What makes complementary angles?
The sum of their degree measure is 180.
Angles that are:
-adjacent
-noncommon sides are opposite rays
Linear Pairs
What makes linear pairs?
Angles that are:
-adjacent
-noncommon sides are opposite rays
What is the vertical angle theorem?
Vertical Angles are congruent
Vertical angles are congruent
Vertical Angle Theorem
If two angles are congruent, what is true about their complements?
Their complements are congruent
If two angles are complementary to the same angle, then they are
Congruent
If two angles are supplementary to the same angle, then they are
Congruent
What is true about two angles that are congruent and supplementary?
They are right angles
What is a theorem? How is it different from a postulate?
A postulate cannot be proven. A theorem can be proven by using logic and deductive reasoning. They are different because one can be proven and the other cannot.
What kinds of statements can you use in a two-column proof?
-definitions
-properties
-postulates
Write a definition for collinear points, and show how the definition can be interpreted as a biconditional.
When one line contains both points
If an only if two points are on one line are they collinear.
Define conjecture in your own words.
A conjecture is an educated guess after an observation; a conclusion from past experiences.