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From Encyclopedia

Postulates, Theorems, and Proofs Postulates, Theorems, and Proofs

Postulates and theorems are the building blocks for proof and deduction in any mathematical system, such as geometry, algebra, or trigonometry. By using postulates to prove theorems, which can then prove further theorems, mathematicians have built entire systems of mathematics. Postulates, or axioms , are the most basic assumptions with which a reasonable person would agree. An example of an axiom is "parallel lines do not intersect." Postulates must be consistent, meaning that one may not contradict another. They are also independent, meaning not one of them can be proved by some combination of the others. There may also be a few undefined terms and definitions. Postulates or axioms can then be used to prove propositions or statements, known as theorems. In doing so, mathematicians must strictly follow agreed-upon rules of argument known as the "logic" of the system. A theorem is not considered true unless it has been rigorously proved by valid arguments that have strictly followed this logic. Deductive reasoning is a method by which mathematicians prove a theorem within the pre-defined system. Deduction begins by using some combination of the undefined terms, definitions, and postulates to prove a first theorem. Once that theorem has been proved by a valid argument, it may then be used to prove other theorems that follow it in the logical development of the system. Perhaps the oldest and most famous deductive system, as well as a paradigm for later deductive systems, is found in a work called the Elements by the ancient Greek mathematician Euclid (c. 300 b.c.e.). The Elements is a massive thirteen-volume work that uses deduction to summarize most of the mathematics known in Euclid's time. Euclid stated five postulates, equivalent to the following, from which to prove theorems that, in turn, proved other theorems. He thereby built his well-known system of geometry: Starting with these five postulates and some "common assumptions," Euclid proceeded rigorously to prove more than 450 propositions (theorems), including some of the most important theorems in mathematics. The Elements is one of the most influential treatises on mathematics ever written because of its unrelenting reliance on deductive proof. Its "postulate-theorem-proof" paradigm has reappeared in the works of some of the greatest mathematicians of all time. What are considered "self-evident truths" may change from one generation to another. Until the nineteenth century, it was believed that the postulates of Euclidean geometry reflected reality as it existed in the physical world. However, by replacing Euclid's fifth postulate with another postulate—"Given a line and a point not on the line, there are at least two lines parallel to the given line"—the Russian mathematician Nikolai Ivanovich Lobachevski (1793–1856) produced a completely consistent geometry that models the space of Albert Einstein's theory of relativity. Thus the modern pure mathematician does not regard postulates as "true" or "false" but is only concerned with whether they are consistent and independent. see also Consistency; Euclid and His Contributions; Proof. Stephen Robinson Moise, Edwin. Elementary Geometry from an Advanced Standpoint. Reading, MA: Addison-Wesley, 1963. Narins, Brigham, ed. World of Mathematics. Detroit: Gale Group, 2001.


From Yahoo Answers

Question:Using AA similarity Postulate, (if two triangles have two congruent angles they are similar) Prove that SSS and SAS prove similarity

Answers:both SSS and SAS prove congruency, and by definition, congruent triangles have all congruent sides and all congruent angles that means that the triangles have 3 congruent angles, and you only need 2 for the AA similarity postulate, so that means SSS and SAS alos both prove similarity please vote best answer:)

Question:Please give a brief (yet concise) explanation/definition of the following postulates and theorems: 1)Segment Addition Postulate 2)Angle Addition Postulate 3)Linear Pair Postulate 4)Corresponding Angle Postulate 5)Alternate Interior Angle Theorem 6)Alternate Exterior Angle Theorem 7)Consecutive Interior Angle Theorem 8)Corresponding Angle Converse 9)Alternate Interior Angle Converse 10)Alternate Exterior Angle Converse 11)Consecutive Interior Angle Converse 12)Congruent Supplements Theorem 13)Congruent Complements Theorem 14)Vertical Angle Theorem 15)Right Angle Theorem 16)Definition of supplementary angles 17)Definition of complementary angles 18)Definition of right angle 19)Definition of perpendicular lines 20)Definition of congruent angles 21)Definition of congruent segments 22)Exterior Angle Theorem 23)Corollary to the Triangle Sum Theorem 24)Third Angle Theorem 25)SSS Postulate 26)SAS Postulate 27)ASA Postulate 28)AAS Theorem 29)HL Theorem 30)Base Angle Theorem Err, yeah... >_>" It's a lot, isn't it?

Answers:Not to be a snot, but seriously, just crack open any geometry text book and you'll get the answers you need.

Question:

Answers:its a math thing about proving the equality of triangles. AAS is angle, angle side--if you have two equivalent angles and a side of the same length then the triangles are equivalent. SSS is side, side, side, if all three sides are the same length, then the triangles are equivalent. SAS is side, angle, side, so if two sides and the angle between them are teh same on the two triangles, then they are equivalent.

Question:

Answers:simple trig? I barely remember back that far but heres your answer enjoy~

From Youtube

4.2 Triangle Congruence by SSS and SAS Part 2 :mrpilarski.wordpress.com Side Side Side Postulate If three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent. SAS Postulate If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.

4.2 Triangle Congruence by SSS and SAS - How to Prove Triangles Congurent :mrpilarski.wordpress.com Side Side Side Postulate If three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent. SAS Postulate If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.