Chap. 2 Math Terms
Law of Detachment
If P --> Q is true, and P is true, then Q is also true
Law of Syllogism
If P --> Q is true, and Q -->R is true, then P --> R is true also
conjunction; ^
compound statement joined by "and" where all statements must be true in order to be true
disjunction
compound statement joined by "or" where all statements have to be false for it to be false
biconditional
conjunction of two statements consisting of a conditional and its converse; false if either the conditional or converse is false
conditional statement
if-then statement, if there are no counter-examples, it is true; P --> Q
hypothesis
phrase immediately following if
conclusion
phrase immediately following then
Right angle theroems
* Perpendicular lines intersect to form four right angles * all right angles are congruent * perpendicular lines form congruent, adjacent angles * if 2 angles are congruent and supplementary, they are right angles * if 2 angles form a linear pair, then they are right angles
Properties of an Equality
* substitution: plug in a variable * reflexive: a=a * symmetric: if a=b, then b=a * transitive: if a=b and b=c, then a=c * addition: if a=b, then a+c = b+c * subtraction: if a=b, then a-c=b-c * multiplication: if a=b, then a * c = b * c * division: if c does not equal zero, then a/c=b/c * distributive: a(b+c)= ab + ac
Postulates (on the justification sheet also
* through any two points there is exactly one line * through any three noncollinear points, there is exactly one plane * A line contains at least two points * a plane contains at least three noncollinear points * if two points like in a plane, then the entire line containing those points lies in that pane * if two lines intersect, then that intersection is a point * if two planes intersect, then their intersection is a line * segment addition postulate * ange addition postulate
steps to writing a proof
1. Rewrite in if -then form 2. Draw - and label a pic according to the if -then statement 3. State - given= hypothesis of conjecture prove= conc. of conjecture (use labels) 4. Plan - in ur head or in discussion 5. Prove - write proof
compound statement
2 or more statements joined by "and" or "or"
Segment Addition Postulate
If a point B lies on a line segment AC, then AB + BC = AC the measure of a line is the sum of the two segments that make it up
Complement Theorem
If the non common sides of 2 adjacent angles form a right angle, then the angles are complementary
Mdpt Theorem
The coordinates of the midpoint of a line segment are the average of the coordinates of its endpoints.
Overlapping Segments Theorem
You are given that AB = CD. Add BC to both sides of the equation, resulting in AB + BC = BC + CD with the Addition prop of equality. AB + BC = AC and BC + CD = BD by the Segment Addition Postulate. use substitution that AC = BD
Which pairs will always have the same truth value
a conditional and a contrapositive, an inverse and a converse
counter-examples
a conjecture based on several observations, it takes one counter-example to show that a conjecture is untrue
inductive reasoning
a pattern of examples or observation to make a conjecture
theorem
a statement or conjecture that has been proven, and it can b used as reason to justify statements in other proofs
conjecture
an educated guess based on known information
Congruent Complements Theorem
angles complementary to the same angle or to congruent angles are congruent
Congruent Supplements Theorem
angles supplementary to the same angle or to congruent angles are congruent
proof
argument to arrive at a conclusion about a geometric statement
converse statement
exchanging the hypothesis and conclusion of the conditional; Q --> P
deductive reasoning
facts, rules, laws, to explain why something is true
Truth Value: Biconditional
false if any part is false
Truth Value: Conjunction
false if any part is false
Vertical Angles Theorem
if 2 angles are vertical angles, then they are congruent
Angle Addition Postulate
if B is in the interior of AOC, then AOB + BOC = AOC the measure of an angle is the sum of the two smaller angels inside
Supplement Theorem
if two angles form a linear pair, then they are supplementary
inverse statement
negating both the hypothesis and conclusion of the conditional; ~P --> ~Q
contrapostive statement
negating both the hypothesis and conclusion of the converse statement; ~Q --> ~P
Truth Value: Disjunction
only false if every part is false
Truth Value: Conditionals
only false when it goes T --> F
the negation; ~
opposite of the original statement
postulate (axiom)
statement that describes a fundamental relationship between the basic terms of geometry
logically equivalent
statements with the same truth value
truth value
whether the statement is true