Chapter 2 Flashcards
Conjecture
A conclusion in a statement form.
Inductive Reasoning
Forming a Conclusion based on examples, history of events, and/or patterns
Counter example
An example the disproves a conjectures
Statement
A statement with a truth value
Truth value
True or false
Negation
Opposite symbol ~
Compound statement
Two state joined together by an and or or
Conjunction
AND compound statement
Disconjuction
OR compound statement
Truth table
A way to organize the truth values of your statements
^ AND
2 or more have to be true
or ⚓️
One or more has to be true
Conditional
“If, then” statement
Converse
If q, then p
Inverse
If ~p, then ~q
Contrapositive
If ~q, then ~p
Biconditional
p if and only if q
Inductive Reasoning
Make a conclusion based on past events and patterns
Deductive Reasoning
Make a conclusion based on rules, laws, theorems, postulates, axiom, and definitions
Law of detachment
Given: p->q
p
Law of Syllogism
Given: p->q
q->r
Conclusion: p->r
Postulate
Rule/law/statement -accepted to be true (not proven)
Theorem
Rule/law/statement - is proven/accepted
Through any 2 points exactly 1 line can be drawn
Postulate 2.1
Through any non-collinear 3 pointy here is exactly 1 plane
Postulate 2.2
A line contains at least 2 points
Postulate 2.3
A plane contains at least 3 non collinear points
Postulate 2.4
If 2 points lie in a plane then all the points that on the line connecting the points lie in the plane
Postulate 2.5
If two lines intersect then their intersection is a point
Postulate 2.6
If two planes intersect then their intersection is a line
Postulate 2.7
If M is the midpoint of AB then AM is congruent to MB
Midpoint of a line segment
Proof
Supporting a conjecture through deductive reasoning
Reflexive property
A=A
Symmetric Property
If x=2 then 2=x
Distributive property
2(x+y)=2x+2y
Transitive property
If x=y and y=z then x=z
Segment Addition Property
AC+CB=AB
Complement theorem
If the noncommon sides of two adjacent angles form a 90 degree angle then the angles are complementary
Congruent Supplements Theorem
If two angles are supplementary to the same angle then these to original angles are equal to eachother
Congruent complements theorem
If two angles are complementary to the same angle then those two angles are equal to eachother
Supplement theorem
If 2 angles form a linear pair, they are supplementary angles
Vertical angles theorem
All vertical angles are congruent
Perpendicular lines form 4 right angles
Right angle theorems
All right angles are congruent
Right angle theorems
Perpendicular lines form congruent adjacent angles
Right angle theorems
If 2 angles are congruent and supplementary then they are right angles
Right angle theorems
If 2 congruent angles form a linear pair then they are right angles
Right angle theorems
Triangle Angle Sum Theorem
The 3 angles of any triangle add to 180
If a point is on the perpendicular bisector of a segment, then it is equidistant from the end point
Perpindicular bisector theorem
Steps to indirect proofs
1) assume temporarily that the conclusion is false
2) reason logically (while giving reasons to support) until a contradiction is reached
3) contradiction has been met then temporary assumption is not valid and the conclusion must be true
The triangle inequality theorem
The sum of any two sides of a triangle must be greater than the third side
If two sides of a triangle are congruent to two sides of another triangle AND the third side of the 1st triangle is greater than the 3rd side of the 2nd triangle, then the angle of the 1st triangle is greater than the angle of the second triangle
Converse of the Hinge Theorem
Sum of all angles of a convex polygon
180(n-2)
Polygon interior angle sum
360 degrees
Each angle of a regular polygon
360/n
Polygon exterior angle sum
Diagonals bisect eachother in a parallelogram
Theorem 6.7
- Both pair of opposite sides are congruent
- both pairs of opposite angles are congruent
- the diagonals bisect eachother
- one pair of opposite sides is congruent and parallel
- has both sides parallel
Tests for parallelograms
Diagonals of a rectangle
Diagonals of a rectangle are congruent
Rhombus has perpendicular diagonals
Theorem 6.15
The diagonals of a rhombus bisect one pair of opposite angles
Theorem 6.16
Rhombus has a pair of consecutive sides that are congruent
Theorem 6.19
A square is a rhombus and a rectangle
Theorem 6.20
Trapezoid
Quad. With one pair of parallel opposite sides
An isosceles trapazoid has each pair of base angles being congruent
Theorem 6.21
If and only if a trapezoid is isosceles then it has congruent diagonals
Theorem 6.23
If a quad is a kite then the diagonals are congruent
Theorem 6.25
If a quad is a kite then exactly one pair of opposite angles are congruent
Theorem 6.26