Transversals and Inequalitities..........................................................

(The Transvesal Postulate) If two parallel lines are cut by a transversal, then corresponding angles have equal measures.

(The Parallel Postulate and Theorem) Given a line and a point not on the line, there is exactly one line through the point that is parallel to the given line.

(The Parallel Postulate) Given a line and a point not on the line, there can be no more than one line through the point that is parallel to the given line.

(The Perpendicular Postulate 1) Given a line and a point not on the line, there can be no more than one line through the point that is perpendicular to the given line.

(The Perpendicular Postulate 2) In a plane, given a line and a point on the line, there is exactly one line through the point that is perpendicular to the given line.

(The Unique Line Postulate) If A and B are two points, then there is exactly one line that contains points A and B (or) Two points determine a line.

(The Unique Point Postulate) If AB is a ray, then there is a unique point P on ray AB such that AP equals any specified length.

(The Line Postulate) A line contains an infinite number of points.

(Whole Greater Than Part Postulate) If a + b = c then c > a and c > b (or) a <> b

(Trichotomy Postulate) If a and b are real numbers, then exactly one of the following statements must be true: a = b or a > b or a < b

(Transitive Postulate of Inequality) If a > b and b > c then a > c (or) a > b > c

(Addition Postulate of Inequality) If a > b then a + c > b + c

Theorems

• Supplements of equal angles are equal.
• Supplements of the same angel are equal.
• Complements of equal angels are equal.
• Complements of the same angel are equal.
• Vertical angels are equal.
• All right angles are equal.
• If two angles form a linear pair and are equal then, each is a right angle.
• The bisector of an angle divides the angle into two angles, each ½ as large as the original angle.
• The bisector of a line segment divides the segment into two segments, each ½ as large as the original line segment.
• If two sides of a triangle have equal lengths, then the angles opposite those sides have equal measures, or the base angles of an isosceles triangle are equal.
• The bisector of the vertex angel of an isosceles triangle separates the triangle into two congruent triangles.
• The bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base of the triangle.
• If two angles of a triangle have equal measures, then the side s opposite those angles have equal lengths.

General Postulates Duducive Proofs................................................................................................

(Substitution Postulate of Equality) A number may be substituted for its equal in an expression.

Example: If A=B then A or B may be substituted for each other.

Therefore, if x = y then x + 2y = 4 then you can replace the (x) with a (y) in the equation

x + 2y = 4 using the substitution postulate of equality to obtain the equation y + 2y = 4 by simplifying we get 3y = 4 and simplification even further gives us the solution that y = 4/3

(Reflexive Postulate of Equality) Any number equals itself.

Example: If A=A then A will always equal A

Therefore, 2 = 2 or 4 = 4

Note: The reflexive postulate is usually used before exercising the subtraction postulate or the transitive postulate.

(Symmetric Postulate of Equality) The members of an equation may be interchanged.

Example: If a = b then b = a
Example: If x + y = 180 then 180 = x + y

Therefore, 90 + 90 = 180 then 180 = 90 + 90

(Transitive Postulate of Equality) Numbers equal to the same number are equal to each other.

Example: If a = b and b = c then a = c

Therefore, if A + B = 180 and C + B = 180
Then, B = B (Reflexive postulate) and
A = C (Transitive postulate)

(Addition Postulate of Equality) If equal numbers are added to equal numbers, the sums are equal.

Example: If a = b and c = d then, a + c = b + d

Therefore, 2 + 2 = 3 + 3 then, 2 + 3 = 2 + 3

(Subtraction Postulate of Equality) If equal numbers are subtracted from equal numbers, the differences are equal.

Example: If a = b and c = d then, a – c = b – d

Therefore, 4 + 4 = 5 + 5 then, 4 – 5 = 4 – 5

(Multiplication Postulate of Equality) If equal numbers are multiplied by equal numbers, the products are equal.

Example: If a = b and c = d then a*c = b*d

Therefore, if ½ A = 45 and 2 = 2
Then, A = 45

(Division Postulate of Equality) If equal numbers are divided by nonzero numbers, the quotients are equal.

Example: If a = b and c = d then a/c = b/d

Therefore, if 2A + 2B =180 and 2 = 2
Then, A + B = 90

(Powers Postulate of Equality) Like powers of equal numbers are equal.

Example: If a = b then, aⁿ = bⁿ

(Roots Postulate of Equality) Like roots of equal numbers are equal.

Example: If a = b then √a = √b