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conditional statement
a logical statement that has two parts: a hypothesis, p and a conclusion, q
example of a conditional statement
if an animal is a monkey, then it has a tail; p: an animal is a money; q: it has a tail; if p, then q
negation of a statement
the opposite of the original statement
example of a negation of a statement
an animal is not a monkey; not p
converse
switch the hypothesis and the conclusion
example of coverse
if an animal has a tail, then it is a monkey; if q, then p
inverse
negate both the hypothesis and the conclusion
example of inverse
if an animal is not a monkey, then it has no tail; if not p, then not q
contrapositive
switch and the negate both the hypothesis and the conclusion
example of contapositive
if an animal has no tail, then it is not a monkey; if not q, then not p
equivalent statements
two statements that are both true or false
example of an equivalent statement
if a figure is a triangle, then it is a polygon
biconditional statement
a statement that contains the phrase “if and only if”; can only be written if both the conditional and converse statements are always true
example of a biconditional statement
a figure is a triangle if and only if it is a polygon; p if and only if q
perpendicular lines
if two lines intersect to form a right angle; their slopes are negative reciprocals
conjecture
an unproven statement based on an observation; an educated guess based on data and examples; is false if there is a counter example
example of a conjecture
prime numbers are odd, proof: 1, 3, 5, 7, 11, 13 are odd
counter example
a specific case for which the conjecture is false
example of a counter example
2 is prime and an even number
inductive reasoning
the process of finding a pattern in specific places and then writing a conjecture for the general case
example of inductive reasoning
most softball players become coaches, sue is a softball player, so she’ll become a coach
deductive reasoning
uses facts, definitions, properties, and the laws of logic (detachment and syllogism) to form a logical argument
example of deductive reasoning
all dogs have ears; golden retrievers are dogs, therefore they have ears
law of detachment
if the hypothesis of true conditional statement is true, then the conclcusion is true
example of the law of detachment
Mary goes to the movies every friday and saturday night, today is a friday; Mary goes to the movies; if pq is true, and p is true, then q is true
law of syllogism
if the statements: hypothesis p, then conclusion q and if hypothesis q, then conclusion r are true then this statement is true: if hypothesis p, then conclusion r; if p = q and q=r, then p=r
example of the law of syllogism
if Jane taks chemistry this year, then Ella will be Janes lab partner, if Ella is Jane’s lab partner, then Jane will get an A is chemistry; if Jane takes chemistry this year, she will get an A in chemistry
two point postulate
through any two points, there, exists exactly one line; if two points, then one line
line point postulate
a line contains at least two points; if line, then two or more points
line intersection postulate
if two lines intersect, then their intersection is one point
three point postulate
through any three non collinear points that exists exactly one plane; if three non collinear points, then one plane
plane point postulate
a plain contains at least three non collinear points; if one plane, then three non collinear points
plane line postulate
if two points lie in a plane, then the line containing them lies in the plane
plane intersection postulate
if two planes intersect, then their intersection is a line
addition property of equality
if a = b, then a + c = b +c
subtraction properties of equality
if a = b, the a-c = b-c
multiplication properties of equality
if a=b, then ac=bc
division property of equality
if a=b and c≠0, then a/c=b
distributive property of equality
for any real numbers a, b, and c; a(b+c)=ab+ac
simplify (combine like terms)
for any real numbers a, b, and x; ax+bx=(a+b)x
symmetric property of equality
if a=b, then b=a
reflexive property of equality
for any real number a: a=a
transitive property of equality
if a=b and b=c, then a=c
substitution property of equality
if a=b, then a could be substituted for b in any expression
proof
a logical statement that uses deductive reasoning to show that a statement is true
two column proofs (left statement)
given information OR the result of applying properties, definitions, or known facts
two column proofs right statement (right statement)
responses for each corresponding statement
reflexive property (properties of segment congruence)
for any segment AB, AB=AB
symmetric property (properties of segment congruence)
if AB is congruent to CD, then Cd is congruent to AB
transitive property (properties of segment congruence)
if AB is congruent to CD, and CD is congruent to EF, then AB is congruent to EF
reflexive property (properties of angle congruence)
For any angle A, measure A is congruent to measure A
symmetric property (properties of angle congruence)
if angle A is congruent to angle B, then angle B is congruent to angle A
transitive property (properties of angle congruence)
if angle A is congruent to angle B, and Angle B is congruent to angle C, then angle A is congruent to angle C
right angles congruence theorem
all right angles are congruent, one right angle is 90*
supplementary angles
two angles that add up to 180*, do not have to be adjacent or share a vertex
congruent supplements theorem
if two angles are supplementary to the same angle (or to congruent angles), then they are congruent
complementary angles
two angles that add up to 90*, do not have to be adjacent or share a vertex
congruent complements theorem
if two angles are complementary to the same angle (or two congruent angles), then they are congruent
linear pair postulate
if two angles form a linear pair, then they are supplementary
vertical angles congruence theorem
vertical angles are congruent