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73 Cards in this Set
- Front
- Back
Parallel
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2 objects are // iff they dont intersect
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Distinct
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2 distinct pts determine a line iff there is one line containing those 2 pts
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concurrent
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2 objects or more meet iff their intersection is nonempty
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Coplanar/collinear
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2 or more pts are collinear iff there is a single place/line containing them
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indirect proof
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A proof by contradiction where you assume p and ~q. Then try to find thr contradiction
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I1
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Any 2 distinct pts determine a line
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I2
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Any 3 noncollinear pts determine a plane but just one plane
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axioms
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Are mathematical statements accepted without proofs
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Distance axioms
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d1-d4
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D1
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For any 2 pts A, B there is a unique real # called DISTANCE from A to B, AB
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triangle inequality
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For any 3 points A, B, C, AB+BC>=AC
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d3
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For any point A, B, AB=BA
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d2
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For any points A, B, AB>=0 and AB=0 iff A=B
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model
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Given a set of axioms, for that set is a mathematical example in which the axioms hold truth
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converse
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Given p-q the converse is q-p
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betweeness
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Given pts A, B, C we say that B is between A & C iff
1) ABC are distinct 2) ABC are collinear 3) AB+BC=AC |
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I4
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If 2 distinct planes meet their intersection is a line
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I3
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If a plane contains 2 pts then it contains every line thru those 2 points
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unprovable
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If a statement fails to be true in any one model for a set of axioms, the statement is unprovable
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Independent
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If for any one axiom there is a model in which that axiom fails but all others hold
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consistent
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If there is no statement that can be proven or disproven
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Conjecture
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if...then
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incidence axioms
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I1-i5
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properties of axioms
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Independence( not redundant)
Consistency (no contradiction) |
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orthocenter
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Intersection of the 3 altitudes of a triangle
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contrapositive
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Is ~q-~p to the p-q
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orthocenter
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Lie on side but not vertex is right
Outside obtuse |
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circumcenter properties
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Midpoint of hypotenus is right triangle
Outside is obtuse Inside is acute |
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Logically equivalent
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P and q are this if p-q and q-p are both true
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circumcenter
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Point where the perpendicular bisectors of a triangle intersect
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Undefined terms
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Point, line, plane, space
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Angel ABC
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Ray BA U Ray BC provided A, B, C are not collinear
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I5
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Space consist of at least 4 noncoplanar pts and at least 3 noncollinear. Every plane contains at least 3 noncol pts abd at least 2 pts
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Direct proof
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Start from p use def, thms to arrive to q
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Proof by contrapositive
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To prove p-q instrad prove ~q-~p (dont assume p, proving p is false)
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incenter
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Where the angle bisectors intersect which is only inside the triangle
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Ray AB
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{X|A-X-B or A-B-X or X=A or X=B}
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Segment AB
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{X|A-X-B or X=A or X=B}
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line AB THM
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if A&B are distinct points then line AB ={X|A-X-B or X-A-B or A-B-X or X=A or X=B}
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segment construction thm
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if segment AB and segment CD are two segments w/AB <CD then there is a point E on segment CD so that C-E-D and CE=AB
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angle ABC
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A, B, C not collinear < ABC =ray AB U ray BC
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a point is interior to an angle
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a point D is interior to angle ABC iff there exists points E on ray BA and F on ray BC neither equal to B so that E-D-F holds
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one Ray is between two others
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ray BD is between rays BA and BC written ray BA - ray BD-ray BC iff these rays are distinct and m< ABD + m<DBC= m<ABC
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two angles form a linear pair
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iff they share one side and other side are opposite rays
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vertical pair thm
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vertical angles have the same measure
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supplementary
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180
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complementary
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90
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a set is convex
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iff whenever A and B are distinct pts in K all segments AB lies in K
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triangle ABC
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given 3 non collinear pts A, B, C triangle ABC is the set =segment AB U segment AC U segment BC
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congruent
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segments AB and XY are congruent AB=XY
Angles 1 and 2 are congruent if m<1 = m<2 triangles abc and xyz are congruent iff angle a=Angle x, b=y, c =z ab=xy, ac=xz, bc=yz |
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SAS
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let triangle abc and xyz be two triangles so that two sides and their included one angle of triangle abc are congruent to the corresponding two sides and angles of triangle xyz under a approach x and b approach y and c approach z then triangle abc is congruent to triangle xyz
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Euler segment
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orthocenter
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incenter
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circumcenter
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a point D is interior
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to <ABC iff there exists pts E on ray BA and F on ray BC neither equal to B so that E-D-F holds
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ray BD is between
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rays BA and BC written RAY BA- RAY BD- ray BC iff these rays are distinct and m<ABD + m<DBC= m<ABC
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a side of a line
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protractor postulate A3
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linear pairs
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two angles form a linear pair iff they share one side and the other side are opposite rays
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opposite rays
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if A-B-C hold then ray BA and Ray Bc are called this
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if abc and bcd hold then abcd holds
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plane separation axiom H1
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triangle abc
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congruence
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SAS
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crossbar thm
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postulate of pasch
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angle construction thm
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segment construction thm
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thm
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thm
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thn of union
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