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Level 154

Axioms & Postulates

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Transitive Axiom
Things equal or congruent to the same or other things, are equal or congruent to each other. If x=y and y=z, then x=z!
Substitution Axiom
A quantity may be substituted for an equal quantity in any expression or equation. If x=y and y=z, then x=z! a+6=10 is equal to a+(2)(3)=10
Partition Axiom
The whole equals the sum of it's parts.
Idenity or Reflexive Axiom
Any quantity (amount) is equal to itself. 64=64
Addition Axiom
If equals are added to equals, the sums are equal. 6+8+2 is equal to 14+2
Subtraction Axiom
If equals are subtracted from equals, the differences are equal.
Multiplication Axiom
If equals are multiplied with equals, the products are equal.
Division Axiom
If equals are divided by equals, the quotients are equal. Alsos, halves of equals are equal.
Powers Axiom
Like powers of equals are equal. If x=y then x2=y2
Roots Axiom
Like roots of equals are equal. if x=y then x1/2=y1/2 (their sq roots are the same)
One Line Postulate
Only one straight line can be drawn between any two points. Line AB is the only straight line between points A and B.
Shortest Distance Postulate
The shortest distance between any two points is the straight line that is drawn between them.
Line has Two points Postulate
Any line contains at least two points.
Two points One line Postulate
through any two points there is one and only one line.
Line Intersection Postulate
two straight lines can intersect each other at one and only one point.
Parallel Postulate
if there is a line and a point not on the line then there is exactly one line through the point parallel to the given line
Perpendicular through line postulate
One and only one perpendicular line can be drawn to or through any point on a line (in a plane).
Perpendicular point to line postulate
One, and only one, perpendicular line can be drawn from or through any point not on a line to that line.
Parallel Transversal postulate
If two parallel lines are cut by a transversal, then corresponding angles are congruent. (ex. p.18)
Midpoint postulate
Any straight line segment will have only one midpoint.
Segment Addition Postulate
If A, B, and C are collinear, then point B is between A and C if and only if AB + BC = AC.
Angle bisector postulate
Any angle has one and only one bisector. (The 2 angles formed by the bisector are congruent)
Angle Addition Postulate
D is in the interior of angle ABC and only if m<ABD + m<DBC = m<ABC.
Arc Sum Postulate
If point B is one arcABC, then the two arcs formed, arcAB and arcBC, sum to the total length of arcABC.
Plane has 3 points postulate
Any plane has at least 3 coplanar noncollinear points.
Three points one plane postulate
Through any three noncollinear points, there is one and only one plane. Also, through any three linear points, there is at least one plane.
Two points and line in plane postulate
If two points lie in a plane, then the line joining them lies in the plane.
Intersection of planes postulate
If two planes intersect, their intersection is a line. (p21)
Four points in space postulate
Space contains at least four points not in the same plane
One circle per radius postulate
One and only one circle can be drawn for a given radial distance r about any center point. (p21)
Change position postulate
Any geometric figure can be moved or relocated to a new position without changing the figure's size or shape.
Area Addition Postulate
(Postulate) The area of a region is the sum of the areas of its non-overlapping parts.
Area congruence postulate
Congruent figures have equal areas
Area square/rectangle postulate
area formula for a square is s2. area formula for a rectangle is bh or lw
SSS triangle congruence postulate
If the 3 sides of one triangle are congruent to the 3 sides of another triange then the triangles are cogruent.
SAS triangle congruence postulate
If two angles and the included angle of one triangle are congruent to the two sides and the included angle of another triangle, then the triangles are congruent.
ASA triangle congruence postulate
Two triangles are congruent if two angles and the included side of one triangle are congruent to the two angles and the included side of the other triangle.
AA similarity triangle postulate
If two angles of the one triangle are congruent to two angles of another triangle, then the triangles are SIMILAR.