Geometry Theorems and Postulates

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bguduguntla  on January 15, 2012

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Geometry Theorems and Postulates

 SMART IDEAS - Symmetric I - Inverse M - Multiplication D - Distributive A - Addition E - Commutative/Inverse R - Reflexive A - Associative T - Transitive
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SMART IDEA S - Symmetric I - Inverse
M - Multiplication D - Distributive
A - Addition E - Commutative/Inverse
R - Reflexive A - Associative
T - Transitive
Postulate 1 - The Distance Postulate To every pair of different points there corresponds a unique positive number
Postulate 2 - The Ruler PostulateThe points of a line can be placed in correspondence with the real numbers in such a way that:
1) to every point of the line corresponds exactly one real number;
2) to every real number there corresponds exactly one point of the line; and
3) the distance between any two points is the absolute value of the difference of the corresponding numbers
Postulate 3 - The Ruler Placement Postulate Given two points P and Q of a line, the coordinate system can be chosen in such a way that the coordinate of P is zero and the coordinate of Q is positive
Postulate 4 - The Line Postulate For every two points there is exactly one line that contains both points
Postulate 5 - The Plane-Space Postulate 1) Every plane contains at least three non collinear points
2) Space contains at least four non coplanar points
Postulate 6 - The Flat Plane Postulate If two points of a line lie in a plane, then the line lies in the same plane
Postulate 7 - The Plane Postulate Any three points lie in at least one plane, and any three non collinear points lie in exactly one plane
Postulate 8 - Intersection of Planes Postulate If two different planes intersect, then their intersection is a line
Postulate 9 - Plane Separation Postulate Given a line and a plane containing it, the points of the plane that do not lie on the line form 2 sets such that:
1) each of the sets is convex
2) If P is in one of the sets and Q is in the other, then the segment PQ intersects the line
Postulate 10 - Space Separation Postulate The points of space are separated into 2 convex sets H1 and H2 by plane P so that if B is an element of H1, Q is an element of H2 then BQ must intersect plane P
Postulate 11 - Angle Measurement Postulate (AMP) For every angle there's a unique measure, 0<X<180
Postulate 12 - Angle Construction Postulate (ACP) In a half plane above a given line AB there's only one ray AC so that the measure of angle BAC = X where X is a specific positive number
Postulate 13 - Angle Addition Postulate (AAP) The whole angle is the sum of its parts
Postulate 14 - Supplementary (Supp) Postulate The angles of a linear pair are supplementary
Postulate 15 - Side Angle Side Postulate (SAS) If in 2 triangles, 2 pairs of corresponding sides are congruent and the including angles are congruent then the triangles are congruent
Postulate 16 - Angle Side Angle Postulate (ASA) If in 2 triangles, 2 pairs of corresponding angles are congruent and the included sides are congruent, then the triangles are congruent
Postulate 17 - Side Side Side Postulate (SSS) If in 2 triangles, all three sides are congruent to corresponding sides then the triangles are congruent
Theorem 2-1 If a-b > 0, then a > b
Theorem 2-2: Def. of Greater Than/Positive Number Sum Property If a = b + c and c > 0, then a > b
Theorem 2-3 Let A, B, and C be points of a line, with coordinates x, y, and z respectively. If x < y < z, then A-B-C
Theorem 2-4 If A, B, and C are three different points of the same line, then exactly one of them is between the other two
Theorem 2-5: Point Plotting Let AB be ray, and let 3 be a positive number. Then there is exactly one point P of AB such that AP = 3
Theorem 2-6: Mid-Point Theorem Every segment has exactly one mid-point
Theorem 3-1 If two different lines intersect, their intersection contains only one point
Theorem 3-2 If a line doesn't lie in a plane, it can only intersect the plane one time
Theorem 3-3 A line and a point not on it determine one plane
Theorem 3-4 2 intersecting lines determine one plane
Theorem 4-1: Transitive of Angle Congruence If Angle A is congruent to Angle B, and Angle B is congruent to Angle C, then Angle A is congruent to Angle C
Theorem 4-2: Congruent Linear Pair If 2 angles are a linear pair and are congruent, then they are right angles
Theorem 4-3: Acute Theorem All complementary angles are acute. If angle 1 is complementary to angle 2 then the measure of anne 1 is less than 90 degrees
Theorem 4-4 All right angles are congruent
Theorem 4-5: Congruent Supplement Theorem If two angles are supplementary and congruent then they are right
Theorem 4-6: Supplement Theorem Supplements of congruent angles are congruent
Theorem 4-7: Complement Theorem Complements of congruent angles are congruent
Theorem 4-8: Vertical Angle Theorem (VAT) Vertical Angles are congruent
Theorem 4-9: Def. of Perpendicular Perpendicular lines form 4 right angles
Theorem 5-1: Segment Equivalence Relation Segment Congruence is: Transitive, Reflexive, Symmetric
Theorem 5-2: Triangle Equivalence Relation Triangle Congruence is: Transitive, Reflexive, Symmetric
Theorem 5-3: Angle Bisector Theorem Every angle has one bisector
Theorem 5-4: Isosceles Triangle Theorem (ITT) If 2 sides of a triangle are congruent, then the angles opposite these sides are congruent
Theorem 5-4.1: Corollary 5-4.1 Every equilateral triangle is equiangular
Theorem 5-5 If two angles of a triangle are congruent, then the sides opposite them are congruent
Theorem 5-5.1: Corollary 5-5.1 Every equiangular triangle is equilateral
Theorem 6-2: Perp. Bisector Theorem (PBT) Points on the perpendicular bisector of AB are equidistant from A and B. Points equidistant from A and B are on the perpendicular bisector of AB
Theorem 6-2.1: Corollary 6-2.1 or PBT Coro Given a segment AB and a line L in the same plane and 2 points of L are each equidistant from points A and B, then L is the perpendicular bisector of AB
Theorem 6-1, 6-2, & 6-3: Unique Perpendicular1) Through a point off a given line, there is at least one line perpendicular to the given line.
2) Through a given external point there is as most one line perpendicular to a given line.
3) In a given plane, through a given line, there is one and only one line perpendicular to a given line, and it proves that these lines from 90 degree angles
Theorem 6-4.1: Corollary 6-4.1 No triangle has 2 right angles
Theorem 7-1: Parts Theorem The whole is bigger than the parts (for a diagram)
Theorem 7-2: Exterior Angle Theorem (EAT) An exterior angle is greater than either remote interior angles
Theorem 7-2.1: Eat Corollary Theorem If a triangle has a right angle, the other angles are acute
Theorem 7-3: Side Angle Angle (SAA) 2 angles and 1 side make triangle congruence
Theorem 7-4: Hypotenuse Leg (HL) This is only for right triangles. If in 2 triangles the corresponding legs and hypotenuses are congruent, then the triangles are congruent
Theorem 7-5: LAOLS The larger angle is opposite the larger side. This is in one triangle. Interchangeable with Theorem 7-6.
Theorem 7-6: LSOLA The larger side is opposite the larger angle. This is in one triangle. Interchangeable with Theorem 7-5.
Theorem 7-7: First Minimum Theorem (FMT) The shortest distance from any point for a line is the perpendicular distance
Theorem 7-8: Triangle Inequality Theorem Any 2 sides of a triangle must exceed the third side
Theorem 7-9: HINGE Given 2 pairs of congruent sides in 2 triangles and unequal included angles, the opposite sides are unequal in the same order as the angles. This is the same as LSOLA only between 2 triangles. Interchangeable with Theorem 7-10.
Theorem 7-10: HINGE Converse Given two pairs of congruent sides in 2 triangles and unequal included sides the opposite angles are unequal in the same order. This is the same as LSOLA only between 2 triangles. Interchangeable with Theorem 7-9.

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