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# Geometry P & T

### Geometry Postulates and Theorems

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|
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.
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.
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.
Postulate 5 Through any two points there exists exactly only one line.
Postulate 6 A line contains at least two points.
Postulate 7 If two intersect, then their intersection is exactly one point.
Postulate 8 Through any three noncollinear points there exists exactly only one plane.
Postulate 9 A plane contains at least three noncollinear points.
Postulate 10 If two points lie in a plane, then the line containing them lies in the plane.
Postulate 11 If two planes intersect then their intersection is a line.
Theorem 2.1 Congruence if Segments Segment congruence is reflexive, symmetric and transitive.
Theorem 2.2 Congruence of Angles Angle congruence is reflexive, symmetric and transitive.
Theorem 2.3 Right Angles Congruence Theorem All right angles are congruent.
Theorem 2.4 Congruent Supplements Theorem If two angles are supplementary to the same angle (or to congruent angles), then they are congruent.
Theorem 2.5- Congruent Complements Theorem If two angles are complementary to the same angle (or to congruent angles), then they are congruent.
Postulate 12- Linear Pair Postulate If two angles form a linear pair, then they are supplementary.
Postulate 16- Corresponding Angles Converse If two lines are cut by a transversal so that corresponding angles are congruent, then the lines are parallel.
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.
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.
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.
Theorem 3.7-Transitive Property of parallel lines If two lines are parallel to the same line then they are parallel to each other.
Postulate 17- Slopes of Parallel Lines In a coordinate plane, two nonveritcal lines are perpendicular are parallel if they have the same slope.
Postulate 18- Slopes pf Perpendicular Lines In a coordinate plane two lines are perpendicular if the product of their slopes is -1.
Theorem 3.8 If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular.
Theorem 3.9 If two lines are perpendicular, then they intersect to form four right angles.
Theorem 3.10 If two sides of two adjacent acute angles are perpendicular, then the angles are complementary.
Theorem 3.11-Perpendicular Transversal Theorem If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other.
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.
Theorem 2.6 Vertical Angles Congruence Theorem Vertical Angles are congruent.
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.
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.
Postulate 15- Corresponding angles postulate If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent.
Theorem 3.1-Alternate Interior Angles Theorem If two lines are cut by a transversal, then the pairs of alternate interior angles are congruent.
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.
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.
Theorem 4.1-Triangle Sum Theorem The sum of the measures of the interior angles of a triangle is 180 degrees.
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.
Corollary to the Triangle Sum Theorem The acute angles of a right triangle are complementary.
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.
Theorem 4.4-Properties of Congruent Triangles Congruent triangles are reflexive, symmetric, and transitive.
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.
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.
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.
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.
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.
Theorem 4.7-Base Angles Theorem If two sides of a triangle are congruent, then the angles opposite them are congruent.
Theorem 4.8-Converse of Base Angles Theorem If two angles of a triangle are congruent, then the sides opposite them are congruent.
Corollary to the Base Angles Theorem If a triangle is equilateral, then it is equiangular.
Corollary to the Converse Base Angles Theorem If a triangle is equiangular, then it is equilateral.
Created by: kathypopovich