Term
| Point-Line-Plane Postulate: Unique Line Assumption |
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Definition
| Through any two points, there is exactly one line. |
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Term
| Point-Line-Plane Postulate: Number Line Assumption |
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Definition
| Every line is a set of points that can be put into a one-to-one correspondence with the real numbers, with any point corresponding to 0 and any other point corresponding to 1. |
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Term
| Point-Line-Plane Postulate: Dimension Assumption |
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Definition
(1) Given a line in a plane, there is a point that is not on the line.
(2) Given a plane in space, there is a point that is not in the plane. |
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Term
| Line Intersection Theorem |
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Definition
| Two different lines intersect in at most one point. |
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Term
| Definition of Parallel Lines |
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Definition
| Two coplanar lines m and n are parallel, written m || n, if and only if they have no points in common or they are identical. |
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Term
| Definition of Line Segment |
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Definition
| The segment with endpoints A and B is the set of points A and B and all points between A and B. |
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Term
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Definition
| The ray with endpoint A and containing point B consists of the points on AB and all points for which B is between it and A. |
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Term
| Definition of Opposite Rays |
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Definition
| Rays AB and AC are opposite rays if and only if A is between B and C. |
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Term
| Distance Postulate: Uniqueness Property |
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Definition
| On a line, there is a unique distance between two points. |
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Term
| Distance Postulate: Distance Formula |
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Definition
| If two points on a line have coordinates x and y, the distance between them is |x - y|. |
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Term
| Distance Postulate: Additive Property |
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Definition
| If B is on segment AC, then AB + BC = AC. |
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Term
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Definition
| The union of two sets A and B is the set of elements which are in A, in B, or in both. |
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Term
| Definition of Intersection |
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Definition
| The intersection of two sets A and B is the set of elements which are in both A and B. |
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Term
| Triangle Inequality Postulate |
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Definition
| The sum of the lengths of any two sides of a triangle is greater than the length of the third side. |
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Term
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Definition
| An angle is the union of two rays that have the same endpoint. |
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Term
| Angle Measure Postulate: Unique Measure Assumption |
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Definition
| Every angle has a unique measure from 0 to 180. |
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Term
| Angle Measure Postulate: Unique Angle Assumption |
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Definition
| Given any ray VA and any real number r between 0 and 180, there is a unique angle BVA in each half-plane of VA such that mBVA = r. |
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Term
| Angle Measure Postulate: Zero Angle Assumption |
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Definition
| If rays VA and VB are the same, then mAVB = 0. |
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Term
| Angle Measure Postulate: Straight Angle Assumption |
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Definition
| If rays VA and VB are opposite, then mAVB = 180. |
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Term
| Angle Measure Postulate: Angle Addition Property |
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Definition
| If ray VC (except for point V) is in the interior of angle AVB, then mAVC + mCVB = mAVB. |
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Term
| Definition of Angle Bisector |
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Definition
| Ray VR is the bisector of angle PVQ if and only if ray VR (except for point V) is in the interior of angle PVQ and mPVR = mRVQ. |
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Term
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Definition
| The degree measure of a minor arc or semicircle AB of circle O is the measure of its central angle AOB. |
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Term
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Definition
| The degree measure of a major arc ACB of circle O is 360 - m(arc)AB. |
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Term
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Definition
| An angle is zero if and only if m = 0. |
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Term
| Definition of Acute Angle |
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Definition
| An angle is acute if and only if 0 < m < 90. |
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Term
| Definition of Right Angle |
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Definition
| An angle is right if and only if m = 90. |
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Term
| Definition of Obtuse Angle |
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Definition
| An angle is obtuse if and only if 90 < m < 180. |
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Term
| Definition of Straight Angle |
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Definition
| An angle is straight if and only if m = 180. |
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Term
| Definition of Complementary Angles |
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Definition
| Two angles are complementary if and only if m1 + m2 = 90. |
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Term
| Definition of Supplementary Angles |
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Definition
| Two angles are supplementary if and only if m1 + m2 = 180. |
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Term
| Definition of Adjacent Angles |
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Definition
| Two non-straight and nonzero angles are adjacent if and only if a common side is interior to the angle formed by the non-common sides. |
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Term
| Definition of Linear Pair |
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Definition
| Two adjacent angles form a linear pair if and only if their non-common sides are opposite rays. |
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Term
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Definition
| If two angles form a linear pair, then they are supplementary. |
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Term
| Definition of Vertical Angles |
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Definition
| Two non-straight angles are vertical if and only if the union of their sides is two lines. |
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Term
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Definition
| If two angles are vertical, then they have equal measures. |
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Term
| Reflexive Property of Equality |
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Definition
| For any real number a, a = a. |
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Term
| Symmetric Property of Equality |
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Definition
| For any real numbers a and b, if a = b, then b = a. |
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Term
| Transitive Property of Equality |
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Definition
| For any real numbers a, b, and c, if a = b and b = c, then a = c. |
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Term
| Addition Property of Equality |
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Definition
| For any real numbers a, b, and c, if a = b, then a + c = b + c. |
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Term
| Multiplication Property of Equality |
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Definition
| For any real numbers a, b, and c, if a = b, then ac = bc. |
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Term
| Corresponding Angles Postulate |
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Definition
| If two corresponding angles have the same measure, then the lines are parallel; if the lines are parallel, then corresponding angles have the same measure. |
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Term
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Definition
| The slope of the line through (x1, y1) and (x2, y2), with x1 ≠ x2, is (y2 - y1) / (x2 - x1). |
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Term
| Parallel Lines and Slopes Theorem |
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Definition
| Two non-vertical lines are parallel if and only if they have the same slope. |
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Term
| Transitivity of Parallelism Theorem |
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Definition
| In a plane, if line l is parallel to line m, and line m is parallel to line n, then line l is parallel to line n. |
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Term
| Definition of Perpendicular Lines |
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Definition
| Two segments, lines, or rays are perpendicular if and only if the lines containing them form a 90-degree angle. |
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Term
| Two Perpendiculars Theorem |
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Definition
| If two coplanar lines l and m are each perpendicular to the same line, then they are parallel to each other. |
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Term
| Perpendicular to Parallels Theorem |
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Definition
| In a plane, if a line is perpendicular to one of two parallel lines, then it is also perpendicular to the other. |
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Term
| Perpendicular Lines and Slopes Theorem |
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Definition
| Two non-vertical lines are perpendicular if and only if the product of their slopes is -1. |
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