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This guide explores key concepts in inductive reasoning, conjectures, and counterexamples. It elucidates how patterns observed in numerical sequences can help us derive the next term, providing examples for clarity. Additionally, it covers fundamental geometric definitions such as points, lines, planes, and their properties including collinearity and coplanarity. Essential postulates regarding intersections of lines and planes are also discussed. This resource is perfect for students looking to enhance their understanding of reasoning and geometry.
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Definitions Chapter 1 Inductive reasoning Conjecture Counterexample
Definitions • Inductive reasoning • Reasoning that is based on patterns you observe Example Find a pattern for each sequence. Use the pattern to find the next term 384, 192, 96, 48, ______
Definitions continued • Conjecture • A conclusion that is made based on inductive reasoning • Counterexample • An example for which a conjecture is incorrect
Definitions • Point – a location (no size) • Line – a set of points that extends in opposite direction without end • Collinear points – points that lie on the same line • Plane – a flat surface with no thickness and extends without end • Coplanar – points that lie in the same plane • Postulate – an excepted statement of fact
Line – or line m Plane –Plane XYZ or Plane R Small dot with capital letter Use any two points or single lower case letter At least 3 noncollinear points or a captial letter Symbols Name
Postulates • 1-1 – Through any two points there is exactly one line • 1-2 – If two lines intersect, then they intersect in exactly one point. • 1-3 – If two planes intersects, then they intersect in exactly one line • 1-4 – Through any three noncollinear points there is exactly one plane
H G E C D A Example 1 Use the diagram above. What is the intersection of plane HGC and plane AED?
Use the diagram to determine the intersection of AD and CD. Example 2 Example 3 Use the diagram below. What is the intersection of plane FBC and plane EHG? is the intersection between the lines
K V JC Definitions • Segment – a part of a line consisting of two endpoints and all points between them • Ray – part of a line that consists of one endpoint and all points of a line to one side of the endpoint • • Segment KV • • C Ray JC J
Definitions • Opposite rays – two collinear rays with the same endpoint • Form a line • • • Ray VK Ray VC C V K Parallel lines – coplanar lines that do not intersect Skew lines – lines that are not parallel and do not intersect
