Plane Geometry: Basic Axioms in Mathematics

Plane Geometry
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Basic Axioms in Mathematics

(1) Quantities that are equal to the same quantity or to equal quantities are equal to each other.
(2 - Addition Axiom) If equals are added to equals, the sums are equal.
(3 - Subtraction Axiom) If equals are subtracted from equals, the remainders are equal.
(4 - Multiplication Axiom) Doubles of equals are equal. In general, if equals are multiplied by equals, the products are equal.
(5 - Divison Axiom) Halves of equals are equal. In general, if equals are divided by equals, the quotients are equal. The divisor must not be zero.
(6) The whole is equal to the sum of all its parts.
(7) The whole is greater than any of its parts.
(8 - Substitution Axiom) A quantity may be substituted for its equal in any process.
(9) If the first of three quantities is greater than the second quantity, and the second is greater than the third, then the first is greater than the third.
(10) If equals are added to, or subtracted from, unequal, the results are unequal in the same order.
(11) If unequal are multiplied or divided by equals, the results are unequal in the same order.
(12) If unequal are added to the unequal of the same order, the sums are unequal in the same order.
(13) If unequal are subtracted from equals, the remainders are unequal in the reverse order.
(14) Like powers or like roots of equal quantities are equal.

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Plane Geometry: Principles in Geometry

Plane Geometry Principles in Geometry (1) A straight line is the shortest line between two points. (2) Only one straight line can be drawn between two points. (3) Two straight lines intersect at only one point. (4) Quantities (Lines) equal to the same quantity (line) are equal to each other. (5) The whole is equal to the sum of all its parts. (6) The whole is greater than any of its parts. (7) If equals are added to equals, then the sums are equal. (8) If equals are subtracted from equals then the remainders are equal. (9) Doubles of equals are equal. (10) Halves of equals are equal. (11) If the first of three quantities is greater than the second, and the second is greater than the third, then the first is greater than the third. (12) All straight lines are equal.

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