Geometry Postulates and Theorems
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Created by:
ChapinSmith on April 27, 2012
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75 terms
Terms | Definitions |
|---|---|
 conditional statement | a logical statement that has two parts: a hypothesis and a conclusion |
| converse | exchange the hypothesis and conclusion |
| inverse | negate both the hypothesis and the conclusion |
| contrapositive | negate the converse |
| biconditional statement | "if and only if" when the converse is also true |
| deductive reasoning | uses facts, definitions, accepted properties, and the laws of logic to form a logical argument |
| inductive reasoning | when you find a pattern in a specific case and then write a conjecture for the general case |
| conjecture | an unproven statement that is based on observations |
| postulate | a rule that is accepted without proof |
| postulate 5 | through any two points there exists one line |
| postulate 6 | a line contains at least two points |
| postulate 7 | if two lines intersect, then their intersection is exactly one point |
| postulate 8 | through any three noncollinear points, there exists exactly 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 plains intersect, then their intersection is a line |
| reflexive property of equality | for any real number a, a=a |
| symmetric property of equality | for any real numbers a and b, if a=b, then b=a |
| transitive property of equality | for any real numbers a, b, and c, if a=b and b=c, then a=c |
| congruence of segments theorem | segment congruence is reflexive, symmetric, and transitive |
| congruence of angles theorem | angle congruence is reflexive, symmetric, and transitive |
| definition of congruent angles | if two angles are congruent, then they are equal |
| right angles congruence theorem | all right angles are congruent |
| congruent supplements theorem | if two angles are supplementary to the same angle, or congruent angles, then they are congruent |
| congruent complements theorem | if two angles are complementary to the same angle, or congruent angles, then they are congruent |
| linear pair postulate | if two angles form a linear pair, then they are supplementary |
| vertical angles congruence theorem | vertical angles are congruent |
| 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 |
| perpendicular postulate | if there is a line and a point not on the line, then there is exactly one line through the point perpendicular to the given line |
| transversal | a line that intersects two or more coplanar lines at different points |
| corresponding angles postulate | if two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent |
| alternate interior angles theorem | if two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent |
| consecutive interior angles theorem | if two parallel lines are cut by a transversal, then the pairs of consecutive interior angles are supplementary |
| corresponding angles converse | if two lines are cut by a transversal so the corresponding angles are congruent, then the lines are parallel |
| alternate interior angles converse | if two lines are cut by a transversal so the alternate interior angles are congruent, then the lines are parallel |
| consecutive interior angles converse | if two lines are cut by a transversal so the consecutive interior angles are supplementary, then the lines are parallel |
| transitive property of parallel lines | if two lines are parallel to the same line, then they are parallel to each other |
| slopes of parallel lines | in a coordinate plane, two nonvertical lines are parallel if and only if they have the same slope |
| slopes of perpendicular lines | in a coordinate plane, two nonvertical lines are perpendicular if and only 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 line 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 |
| perpendicular transversal theorem | if a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other |
| lines perpendicular to a transversal theorem | in a plane, if two lines are perpendicular to the same line, then they are parallel to eachother |
| triangle sum theorem | the sum of the measures of the interior angles of a triangle is 180 degrees |
| exterior angles theorem | the measure of an exterior angles 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 |
| SSS Congruence Postulate | all three sides of a triangle are congruent |
| SAS Congruence Postulate | two sides and the included angle of a triangle are congruent |
| HL Congruence Theorem | the hypotenuse and one of the legs of a triangle are congruent |
| ASA Congruence Theorem | two angles are the included side of a triangle are congruent |
| AAS Congruence Theorem | two angles and a non-included side of a triangle are congruent |
| base angles theorem | if two sides of a triangle are congruent, then the angles opposite them are congruent |
| converse of base triangles 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 of base angles theorem | if a triangle is equiangular, then it is equilateral |
| midsegment theorem | the segment of connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long as that side |
| perpendicular bisector theorem | in a plane, if a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment |
| converse of the perpendicular bisector theorem | in a plane, if a point is equidistant form the endpoints of a segment, the it is on the perpendicular bisector of the segment |
| concurrency of perpendicular bisectors of a triangle | the perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle |
| concurrent | when three or more lines, rays, or segments intersect in the same point, they are_________ lines, rays or segments |
| angle bisector theorem | if a point is on the bisector of an angle, then it is equidistant from the two sides of the angle |
| converse of the angle bisector theorem | if a point is in the interior of an angle and is equidistant from the sides of the angle, then it lies on the bisector of the angle |
| incenter | the point of concurrency of the three angle bisectors of a triangle |
| angle bisector | a ray that divides an angle into two congruent adjacent angles |
| concurrency of medians of a triangle | the medians of a triangle intersect at a point that is two thirds the distance from each vertex to the midpoint of the opposite end |
| centroid | the point of concurrency in a triangle |
| median of a triangle | a segment from a vertex to the midpoint of the opposite side |
| concurrency of altitudes of a triangle | the lines containing the altitudes of a triangle are concurrent |
| orthocenter | the point at which the lines containing the three altitudes of a triangle intersect |
| theorem 5.10 | if one side of a triangle is longer than another side, then the angle opposite the longer side is larger than the angle opposite the shorter side |
| theorem 5.11 | if one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle |
| triangle inequality theorem | the sum of the lengths of any two sides of a triangle is greater than the length of the third side |
| hinge theorem | if two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first is longer than the third side of the second |
| converse of hinge theorem | if two sides of one triangle are congruent to two sides of another triangle, and the third side of the first is longer than the third side of the second, then the included angle of the first is larger than the included angle of the second |
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