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Transcript
Chapter 10: Geometry Section 10.1: Visualization Important Objects Object Visualization Characteristics Point A tiny dot Has no size or shape Line An infinitely long stretched string with no beginning or end An infinite flat piece of paper with no beginning or end Has no thickness Plane Has no thickness Line segment: part of a line lying between two points, called the endpoints of that segment Ray: part of a line lying on one side of a point Where objects lie • In the plane: on an infinite 2-dimensional surface • In space: in an infinite 3-dimensional room Section 10.2: Angles • Def: (a) An angle is the amount of rotation about a fixed point (b) An angle is the region between two rays with a common endpoint • Def: Two angles are congruent if they both represent the same amount of rotation Measuring Angles • Def: Angles are measured in degrees, where a full circle rotation is considered to be 360˚. • Def’s: acute angle: < 90˚ right angle: = 90˚ obtuse angle: > 90˚ straight angle: = 180˚ Angles formed by two lines • Theorem 1: When two lines meet in a plane, they form four angles, which sum to 360˚. • Def: If the angles formed by two intersecting lines are all 90˚, then the two lines are perpendicular. • Def: If two lines never meet, they are parallel. Configurations of 3 lines in a plane • Parallel Postulate (or Axiom): If parallel lines are cut by another line, then the corresponding angles are equal, i.e. 𝑎 = 𝑎′ , 𝑏 = 𝑏 ′ , 𝑐 = 𝑐 ′ , and 𝑑 = 𝑑′ . Configurations of 3 lines in a plane • Converse of the Parallel Postulate: If the angles 𝑎 and 𝑎′ are congruent when two lines 𝑙1 and 𝑙2 are crossed by a third line, then the lines 𝑙1 and 𝑙2 are parallel, or 𝑙1 ǁ 𝑙2 . Configurations of 3 lines in a plane • Theorem 2: The three angles in a triangle sum to 180˚.