September 24, 2010 Foundations of Geometry Important Vocabulary Undefined Terms: a term that is used without a specific example or description. We use these terms to describe other terms in geometry. 1) Point: a location. A point has neither shape nor size. When we name a point, we name it by a capital letter. Example: W 2) Line: made up of points and has no thickness or width. There is exactly one line through any two points. A line can either be named by two points on that line or a lowercase script letter. Example: k X Y 3) Plane: a flat surface that extends indefinitely in all directions. There is exactly one plane through any three points that are not on the same line. A plane is named by a capital script letter or by any three points on that plane. Example: A W X Y September 24, 2010 Fundamental Definitions in Geometry Definitions Related to Points: · Space: the set of all points. · Figure: any set of points. · Plane Figure: all points lie on the same plane. · Space Figure: a 3-D figure, sketched beyond the plane. · Intersection: the set of all points common to two or more figures. · Collinear Points: points that lie on the same line. · Non-Collinear Points: points that do not lie on the same line. · Coplanar Figures: figures that lie on the same plane. · Non-Coplanar Figures: figues that do not lie on the same plane. Definitions Related to Line Segments: 1) Line Segment (segment): a part of a line that begins at one point and ends at another. Example: A B 2) Length of a Segment: distance of a segment between its two endpoints. 3) Congruent Segments: segments that have the same exact distance. 5 cm Example: A B 5 cm C D 4) Midpoint: divides a segment into two congruent segments. 2 in 2 in Example: C M D September 24, 2010 Basic Postulates for Points, Lines, and Planes Postulates: statements that are considered true, even without proof. · Unique Line Postulate: through any two points is exactly one line. · Line Intersection Postulate: if lines intersect, then they intersect in exactly one point. · Plane Intersection Postulate: through any three non-collinear points, there is exactly one plane. Definitions Related to Rays 1) Ray: a part of a line that begins at one point and extends infinitely (without end) in one direction. We name a ray by its endpoint and one other point on that ray. Example: A J 2) Endpoint of a Ray: the point at which the ray extends from. 3) Opposite Rays: two rays that meet together to form a line. Opposite rays share a common endpoint and extend in opposite directions. Example: A B C 4) Angle: a figure formed by two rays with a common endpoint. Each ray is the side of the angle and each endpoint is thevertex of the angle. ***A vertex of an angle is the place where the two sides of the angle meet*** W Example: X Y September 24, 2010 Definitions Related to Angle Measure 1) Congruent Angles: Angles that are equal in measure. A Example: T C E O M 2) Acute Angle: an angle whose measure is greater than 0 degrees, but less than 90 degrees. Example: L Y N 3) Obtuse Angle: an angle whose measure is greater than 90 degrees, but less than 180 degrees. Example: N V I 4) Right Angle: an angle whose measure is exactly 90 degrees. O Example: M A 5) Straight Angle: an angle whose measure is exactly 180 degrees. Example: P A T Angle Pairs 1) Complementary Angles: Two angles whose measure has a sum of 90 degrees. Each angle is said to be the complement of each other. Example: Q T R S 2) Supplementary Angles: Two angles whose measure has a sum of 180 degrees. Each angle is said to be the supplement of each other. Example: W D R E 3) Adjacent Angles: Two coplanar angles that share a common side and a comment vertex, but have no interior points. Example: N K ο 51 ο 28 I C 4) Linear Pair: When the two uncommon sides of adjacent angles are opposite rays. A linear pair creates a line. Example: Y A N D 5) Linear Pair Postulate: If two angles form a linear pair, then they are supplementary. 6) Vertical Angles: Two angles that are directly opposite each other. A pair of vertical angles are congruent. Examples: S T E <SEP & <TEH are vertical angles P H September 24, 2010 Different Types of Lines 1) Parallel Lines: coplanar lines that never intersect. The symbol for parallel lines is: || Example: k l 2) Perpendicular Lines: coplanar lines that intersect to form right angles. The symbol for perpendicular lines is: ┴ m Example: h 3) Skew Lines: lines that are not on the same plane. Example:.