Geometry Theorems and Postulates
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Created by:
bguduguntla on January 15, 2012
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Terms | Definitions |
|---|---|
 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 Postulate | The 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 Perpendicular | 1) 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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