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# Theorems and Postulates

 Ruler Postulate1) The points on a line can be paired with the real numbers in such a way that any two points can have coordinates 0 and 1 2) Once a coordinate system has been chosen in this way, the distance between any 2 points equals the absolute value of the difference of their coordinates.
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#### Definitions

Ruler Postulate 1) The points on a line can be paired with the real numbers in such a way that any two points can have coordinates 0 and 1
2) Once a coordinate system has been chosen in this way, the distance between any 2 points equals the absolute value of the difference of their coordinates.
Segment Addition Postulate If B is between A and C, then AB + BC = AC
Protractor PostulateOn line AB in a given plane, choose any point O between A and B. Consider →OA and →OB and all the rays that can be drawn from O on one side of line AB. These rays can be paired with the real numbers from 0 to 180 in such a way that: a) →OA is paired with 0, and →OB with 180. b) If →OP is paired with x, and →OQ with y, then m∠POQ = |x - y|.
Angle Addition Postulate 1) If point B lies in the interior of <AOC then m<AOB+m<BOC=m<AOC
2) If <AOC is a straight angle and B is any point not on line AC, then m<AOB + m<BOC =180
Postulate 5 A line contains at least 2 points; a plane contains at least 3 points not all in 1 line; space contains at least 4 points not all in 1 plane.
Postulate 6 Through any 2 points there is exactly 1 line.
Postulate 7 Through any 3 points there is at least one plane, and through any 3 non-collinear points there is exactly 1 plane.
Postulate 8 If 2 points are in a plane, then the line that contains those points is in that plane.
Postulate 9 If 2 planes intersect, then their intersection is a line.
Theorem 1-1 If 2 lines intersect, then they intersect in exactly 1 point.
Theorem 1-2 Through a line and a point not in the line there is exactly 1 plane.
Theorem 1-3 If 2 lines intersect, then exactly 1 plane contains the lines.
Theorem 2-1 the midpoint theorem If M is the midpoint of line AB, then AM = 1/2(AB) and MB = 1/2(AB)
Theorem 2-2 the angle bisector theorem If line segment BX is the bisector of angle ABC, then m<ABX = 1/2(m<ABC) and m<XBC = 1/2 m<ABC
Theorem 2-3 vertical angles are congruent
Theorem 2-4 If 2 lines are perpendicular, then they form congruent, adjacent angles.
Theorem 2-5 If 2 lines form congruent, adjacent angles, then the lines are perpendicular.
Theorem 2-6 If the exterior sides of 2 adjacent, acute angles are perpendicular, then the angles are complementary
Theorem 2-7 If 2 angles are supplements of congruent angles (or of the same angle), then the 2 angles are congruent.
Theorem 2-8 If 2 angles are complements, of congruent angles (or of the same angle), then the 2 angles are congruent.
Postulate 10 If 2 parallel lines are cut by a transversal, then corresponding angles are congruent.
Theorem 3-2 If 2 parallel lines are cut by a transversal, then alternate interior angles are congruent.
Theorem 3-3 If 2 parallel lines are cut by a transversal, then same side interior angles are supplementary.
Theorem 3-4 If a transversal is perpendicular to 1 of 2 parallel lines, then it is perpendicular to the other 1 also.
Postulate 10 If 2 lines are cut by a transversal and corresponding angles are congruent, then the lines are parallel.
Theorem 3-5 If 2 lines are cut by a transversal and the alt. int. angles are congruent, then the lines are parallel.
Theorem 3-6 If 2 lines are cut by a transversal and same-side int. angles are supplementary, then the lines are parallel.
Theorem 3-7 In a plane, 2 lines perpendicular to the same line are parallel.
Theorem 3-8 Through a point outside a line, there is exactly 1 line parallel to the given line.
Theorem 3-9 Through a point outside a line, there is exactly 1 line perpendicular to the given line.
Theorem 3-10 2 lines parallel to a 3rd line are parallel to each other.
Theorem 3-11 The sum of the measures of a triangle is 180.
Corollary 1 If 2 angles of 1 triangle are congruent to 2 angles of another triangle, then the 3rd angles are congruent.
Corollary 2 Each angle of an equiangular triangle has a measure of 60.
Corollary 3 In a triangle, there can be at most 1 right or obtuse angle.
Corollary 4 The acute angles of a right triangle are complementary.
Theorem 3-12 The measure of an exterior angle of a triangle = the sum of the measures of the 2remote interior angles.
Theorem 3-13 The sum of the measures of the angles of a convex polygon with n sides is (n-2)180
Theorem 3-14 The sum of the measures of the exterior angles of any convex polygon, one angle at each vertex, is 360
Postulate 12 - SSS Postulate If 3 sides of 1 triangle are congruent to 3 sides of another triangle, then the triangles are congruent.
Postulate 13 - SAS Postulate If 2 sides and the included angle of 1 triangle are congruent to 2 sides and the included angle of another triangle, then the triangles are congruent
Postulate 14 - ASA Postulate If 2 angles and the included side of 1 triangle are congruent to 2 angles and the included side of another triangle, the the triangles are congruent.
Theorem 4-1 Isosceles Triangle Theorem If 2 sides of a triangle are congruent, then the angles opposite those sides are congruent.
Corollary 1 An equilateral triangle is also equiangular
Corollary 2 An equilateral triangle has 3 60-degree angles
Corollary 3 The bisector of the vertex angle of an isosceles triangle is perpendicular to the base at its midpoint
Theorem 4-2 If two angles of a triangle are congruent, then the sides opposite those angles are congruent
Corollary of Theorem 4-2 An equiangular triangle is also equilateral
Theorem 4-3 AAS Theorem If two angles and a non-included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent.
Theorem 4-4 HL Theorem If the hypotenuse and a leg of one right triangle are congruent to the corresponding parts of another right triangle, then the triangles are congruent.

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