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Level 155

Axioms & Postulates II


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Vertical Angles
Angles opposite one another when two lines intersect.
a bisector
divides an angle into two congruent angles
a midpoint
divides a line into two congruent segments
a rectangle has
four right angles and two pairs of parallel lines
a parallelogram has
two pairs of parallel sides
a square has
four right angles, four congruent sides, and two pairs of parallel sides
a rhombus has
four congruent sides, and two pairs of parallel sides
a trapezoid has
one pair of parallel sides
corresponding angles are congruent
if two parallel lines are cut by a transversal then
Congruent
A plane figure with the same size and shape.
Supplementary
Angles that add to 180 degrees.
Complementary
Two angles whose sum is 90 degrees.
180°
Degrees in a line
A+B>C
if a triangle has sides A, B, & C, and A <_ B <_ C <_, then
sides, angles
a regular polygon has all_______ congruent and all ________ congruent
360°
Degrees in a circle
Hypotenuse
in a right triangle, the side opposite the right angle
Isosceles
two sides have the same length
perpendicular
two lines that intersect and form a right angle are
parallel
lines in the plane that are the same distance apart
B=A
the property of symmetry: if A=B then
a
the reflexive property: A=
c
the transitive property: if A=B and B=C, then A=
Length Postulate
To every segment we can assign a positive number called length that represents how many times a chosen unit length (meter, centimeter, inch, etc.) fits into the segment.
Measures Addition Postulate
We can add and subtract measures (lengths of segments).
Parts-whole Postulate
The whole is equal to the sum of its parts.
Angle Measurement Postulate
To every angle we can assign a non-negative number that represents how many times a chosen angle unit fits into the measured angle.
Parts-whole Postulate (angle addition)
The sum of the parts is equal to the whole.
Postulate about two lines and transversal
If two lines are intersected by a transversal so that the alternate interior (exterior) angles are congruent, then the lines are parallel.
Parallel Line Postulate
Through a point not on a given line, there is only one line parallel to the given line.
Postulate about two parallel lines and transversal
If two parallel lines are intersected by a transversal, then their alternate interior (exterior) angles are congruent.
Existence Axiom
The collection of all points forms a nonempty set. There is more than one point in the set.
Incidence Axiom
Every line is a set of points. For every pair of distinct points A and B there is exactly one line l such that A on l and B on l.
Ruler Placement Postulate
For every pair of distinct points P and Q, there is a coordinate function f: PQ -> R such that f(P)=0 and f(Q)>0.
Plane Separation Postulate
For every line l, the points that do not lie of l form two disjoint, nonempty sets H1 and H2, called half-planes bounded by l, such that
Protractor Postulate
Given any angle, the measure can be put into one-to-one correspondence with real numbers between 0 and 180.
Side Angle Side Postulate
If tri ABC and tri DEF are two triangles such that AB congruent to DE, <ABC congruent to <DEF, and BC congruent to EF, then tri ABC is congruent to tri DEF
Reflection Postulate
For every line l there exists a transformation rho.l: P->P, called the reflection in l such that
Euclidean Parallel Postulate
For every line l and for every external point P, there is exactly one line m such that P lies on m and m//l.
Elliptic Parallel Postulate
For every line l and for every external point P, there is no line m such that P lies on m and m//l.
Hyperbolic Parallel Postulate
For every line l and for every external point P, there are at least two lines m and n such that P lies on both m, n and m//l, n//l.