Geometry Postulates and Theorems
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| Postulate 1-Ruler Postulate | The distance between points A and B is the absolute value of the distance of the coordinates of A and B. AB=|x2-x1|
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| Postulate 2-Segment Addition Postulate | If B is between A and C, then AB+BC=AC. If AB+BC=AC, the B is between A and C.
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| Postulate 3-Protractor Postulate | The measure of angle AOB is equal to the absolute value of the difference between the real numbers for ray OA and ray OB.
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| Postulate 4-Angle Addition Postulate | If P is in the interior of angle RST, then measure of angle RST= measure of angle RSP+ measure of angle PST.
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| Postulate 5 | Through any two points there exists exactly only one line.
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| Postulate 6 | A line contains at least two points.
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| Postulate 7 | If two intersect, then their intersection is exactly one point.
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| Postulate 8 | Through any three noncollinear points there exists exactly only one plane.
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| Postulate 9 | A plane contains at least three noncollinear points.
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| Postulate 10 | If two points lie in a plane, then the line containing them lies in the plane.
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| Postulate 11 | If two planes intersect then their intersection is a line.
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| Theorem 2.1 Congruence if Segments | Segment congruence is reflexive, symmetric and transitive.
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| Theorem 2.2 Congruence of Angles | Angle congruence is reflexive, symmetric and transitive.
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| Theorem 2.3 Right Angles Congruence Theorem | All right angles are congruent.
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| Theorem 2.4 Congruent Supplements Theorem | If two angles are supplementary to the same angle (or to congruent angles), then they are congruent.
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| Theorem 2.5- Congruent Complements Theorem | If two angles are complementary to the same angle (or to congruent angles), then they are congruent.
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| Postulate 12- Linear Pair Postulate | If two angles form a linear pair, then they are supplementary.
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| Postulate 16- Corresponding Angles Converse | If two lines are cut by a transversal so that corresponding angles are congruent, then the lines are parallel.
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| Theorem 3.4- Alternate Interior Angles Converse | If two lines are cut by a transversal so the alternate interior angles are congruent, then the lines are parallel.
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| Theorem 3.5 Alternate Exterior Angles Converse | If two lines are cut by a transversal so the alternate exterior angles are congruent, then the lines are parallel.
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| Theorem 3.6- Consecutive Interior Angles Converse | If two lines are cut by a transversal so the consecutive interior angles are supplementary, then the lines are parallel.
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| Theorem 3.7-Transitive Property of parallel lines | If two lines are parallel to the same line then they are parallel to each other.
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| Postulate 17- Slopes of Parallel Lines | In a coordinate plane, two nonveritcal lines are perpendicular are parallel if they have the same slope.
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| Postulate 18- Slopes pf Perpendicular Lines | In a coordinate plane two lines are perpendicular if the product of their slopes is -1.
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| Theorem 3.8 | If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular.
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| Theorem 3.9 | If two lines are perpendicular, then they intersect to form four right angles.
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| Theorem 3.10 | If two sides of two adjacent acute angles are perpendicular, then the angles are complementary.
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| Theorem 3.11-Perpendicular Transversal Theorem | If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other.
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| Theorem 3.12- Lines Perpendicular to a Transversal Theorem | In a plane, if two lines are perpendicular to the same line, then they are parallel to each other.
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| Theorem 2.6 Vertical Angles Congruence Theorem | Vertical Angles are congruent.
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| Postulate 13- Parallel Postulate | If there is a line and a point not on the line, then there is exactly one line through the point parallel to the given line.
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| Postulate 14- Perpendicular Postulate | If there is a line and point not on the line, then there is exactly one line through the point perpendicular to the given line.
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| Postulate 15- Corresponding angles postulate | If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.
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| Theorem 3.1-Alternate Interior Angles Theorem | If two lines are cut by a transversal, then the pairs of alternate interior angles are congruent.
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| Theorem 3.2 Alternate Exterior Angles Theorem | If two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent.
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| Theorem 3.3-Consecutive Interior Angles Theorem | If two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary.
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| Theorem 4.1-Triangle Sum Theorem | The sum of the measures of the interior angles of a triangle is 180 degrees.
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| Theorem 4.2-Exterior Angle Theorem | The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles.
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| Corollary to the Triangle Sum Theorem | The acute angles of a right triangle are complementary.
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| Theorem 4.3-Third Angles Theorem | If two angles of one triangle are congruent to two angles of a another triangle, then the third angles are also congruent.
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| Theorem 4.4-Properties of Congruent Triangles | Congruent triangles are reflexive, symmetric, and transitive.
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| Postulate 19-Side-Side-Side (SSS) Congruence Postulate | If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent.
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| Postulate 20-Side-Angle-Side (SAS) Congruence Postulate | If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent.
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| Theorem 4.5-Hypotenuse-Leg Congruence Theorem | If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second triangle, then the two triangles are congruent.
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| Postulate 21-Angle-Side-Angle (ASA) Congruence Postulate | If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the two triangles are congruent.
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| Theorem 4.6-Angle-Angle-Side (AAS) Congruence Theorem | If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of a second triangle, then the two triangles are congruent.
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| Theorem 4.7-Base Angles Theorem | If two sides of a triangle are congruent, then the angles opposite them are congruent.
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| Theorem 4.8-Converse of Base Angles Theorem | If two angles of a triangle are congruent, then the sides opposite them are congruent.
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| Corollary to the Base Angles Theorem | If a triangle is equilateral, then it is equiangular.
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| Corollary to the Converse Base Angles Theorem | If a triangle is equiangular, then it is equilateral.
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