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Postulates are statements that are assumed true. Postulates of Euclidean Geometry Point-Line-Plane Postulate: a. unique line assumption: through any two points there is exactly one line b. dimension assumption: given a line in a plane, there exists a point in the plane not on the line c. number line assumption: every line is a set of points that can be put in a one-to-one correspondence with real numbers, with any point on it corresponding to zero and any other point corresponding to one d. distance assumption: on a number line, there is a unique distance between two points Arithmatic and Algebra Postulates Postulates of Equality: Reflexive Property of Equality: a = a Symmetric Property of Equality: if a = b, then b = a Transitive Property of Equality: if a = b and b = c, then a = c Postulates of Equality and Operations: Addition Property of Equality: if a = b, then a + c = b + c Multiplication Property of Equality: if a = b, then ac = bc Substitution Property of Equality: if a = b, then a may be substituted for b in any expression Posulates of Inequality and Operations: Addition Property of Inequality: if a < b, then a + c < b + c Multiplication Property of Inequality: if a < b and c > 0, then ac < bc if a < b and c < 0, then ac > bc Equation to Inequality Property: if a and b are positive and a + b = c, then c > a and c > b Transitive Property of Inequality: if a < b and b < c, then a < c Postulates of Operations: Communtative Property of Addition: a + b = b + a Communtative Property of Multiplication: ab = ba Distributive Property: a(b+c) = ab + ac Triangle Inequality Postulate: the sum of the lengths of two sides of any triangle is greater than the length of the 3rd side Angle Measure Postulate: a. unique measure assumption: every angle has a unique measure from 0 to 180 degrees b. two sides of line assumption: given any ray VA and any number x between 0 and 180 there are unique rays VB and VC such that segment BC intersects line VA and measure of angle BVA is equal to the measure of angle CVA that equals x c. Zero Angle Assumption: if ray VA and ray VB are the same ray, then the measure of angle AVB equals 0 d. Straight Angle Assumption: if ray VA and ray VB are opposite rays, then the measure of angle AVB equals 180 e. Angle Addition Property: if ray VC (except for point V) is in the interior of angle AVB, then the measure of angle AVC + the measure of angle CVB = the measure of angle AVB Corresponding Angles Postulate: if two coplaner lines are cut by a transversal so that two corresponding angles have the same measure, then the lines are parallel Parallel Lines Postulate: if two lines are parallel and cut by a transversal corresponding angles have the same measure Reflection Postulate: Under a Reflection: a. There is a one-one correspondence between points and their images b. If three points are collinear, then their images are collinear c. If B is between A and C, then the image of B is between images of A and C d. The distance between two preimages equals the distance between their images e. The image of an angle is an angle of the same measure f. A polygon and its image, with vertices taken in corresponding order, have opposite orientations Area Postulate: a. Uniqueness Property: given a unit region, every polygonal region has a unique area b. Rectangle Formula: the area of a rectangle with dimensions l and w lw c. Congruence Property: congruent figures have the same area d. Additive Property: the area of the union of two non-overlapping regions is the sum of the areas of the regions Volume Postulate: a. Uniqueness Property: given a unit cube, every polyhedral solid has unique volume b. Box Volume Formula: the volume of a box with dimensions l,w, and h is lwh c. Congruence Property: congruent figures have the same volume d. Additive Property: the volume of the union of two non-overlapping solids is the sum of the volumes of the solids e. Cavalieri's Principle: let I and II be solids included between parallel planes. If every place p parallel to the given planes intersects I and II in sections with the same area then volume (I) equals volume (II) Postulates of Logic: Law of Detachment: If you have a statement or given information p and the justification of the form if p, then q, you may conclude q Law of Transtitivy: If if p, then q and if q, then r, then if p, then r Law of the Contrapositive: A statement if p, then q and its contrapositive if not q, then not p are either both true or both false Law of ruling out Possibilities: when p or q is true and q is not true then p is true Law of inderect reasoning: if reasoning from a statement p leads to a false conclusion, then p is false Postulates of Euclid: 1. Two points determine a line segment 2. A line segment can be extended indefinitely along a line 3. A circle can be drawn with any center and any radius 4. All right angles are congruent 5. If two lines are cut by a transversal, then the interior angles on the same side of the transversl have a total measure less than 180, then the lines will intersect on that side of the transversal