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Section 1.1

Section 1.1. Patterns and Inductive Reasoning Points, Lines, and Planes. Objectives. Find and describe patterns Use inductive reasoning to make real life conjectures Understand and use the basic undefined and defined terms in geometry Sketch the intersection of lines and planes.

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Section 1.1

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  1. Section 1.1 • Patterns and Inductive Reasoning • Points, Lines, and Planes Objectives • Find and describe patterns • Use inductive reasoning to make real life conjectures • Understand and use the basic undefined and defined • terms in geometry • Sketch the intersection of lines and planes

  2. Conjecture – an unproven statement that is based on observations Inductive Reasoning – process of looking for patterns and making conjectures • To prove a conjecture istrue – you must prove that it is • true for all cases. • To prove a conjecture is false – you must prove it is false • for just one case. Example: Conjecture: If you are taking geometry, then you must be a sophomore True or False False– could be a junior. Counterexample – an example that shows a conjecture is false.

  3. Undefined Terms: Terms which do not have a formal definition, but there is an agreement about what they mean. Example: Point , line and plane Defined Terms: Terms which do have a formal definition. Example: Line segment, endpoints, ray, opposite rays and intersection

  4. Point: A • Has no dimension • Represents a location in space • Represented by a small dot • Named using a capital letter Point A Line: c A B • Has one dimension - length • Extends infinitely in opposite directions • Named using two points on the line or • scripted letter Or line c Symbol: or Name the following line in as many ways as possible: B L U E

  5. Line segment or segment: AB AC AD BC BD CD • Has one dimension - length • Is part of a line • Has endpoints – a beginning and an end • Named using its endpoints Q R Symbol: or Name each of the segments in the following figure: A B C D

  6. Is AM the same as RM? Ray • Has one dimension - length • Is part of a line • Has one endpoint • Extends infinitely in one direction • Named using its endpoint first and then • another point on the ray F B Symbol: What are all the ways that the whole ray can be named? A R M Y No – different starting point

  7. Are BC and CB opposite rays? BA and BC are opposite rays Opposite Rays A B C • Rays which share the same endpoint • and extend in opposite directions • Any two opposite rays are collinear No – different starting points Collinear points – Points that lie on the same line Noncollinear points – Points that do not lie on the same line

  8. M B C A Plane • Extends in two dimensions • Is a flat surface • Extends without end • Named using 3 non-collinear • or a scripted letter Plane ABC or plane M Coplanar points – Points that lie on the same plane Noncoplanar points – Points that do not lie on the same plane

  9. A B G H F I C D E Why do we need to use three non- collinear points when naming a plane? Place your pencil on Plane ABH. Top of the box. Place your pencil on Plane BHC. Right side of the box. Place your pencil on Plane FGH. Is it the top or the front? Place your pencil on Plane HID. Is it the front or the right side?

  10. Line 1 Line 2 A B C D E Plane DCB and Plane CBF create CB Plane DCG and Plane EFG create HG F H G Intersect • Two or more geometric figures intersect if they have one or more points • in common • Where figures come together • What is created by the intersection of two lines? A Point • What is created when two planes intersect? A Line • Name two planes that intersect and the line that • their intersection makes.

  11. Is it possible for two planes not to intersect? If yes, name two that do not intersect. A B C D E F H G Intersection (continued) • What is created when three planes • intersect each other? Planes ABC, BCF, and GDH intersect at…? Point C Plane EFG and Plane ABC Plane ADH and Plane FGC

  12. A B C D E F H G Parallel Lines: Two or more coplaner lines which never intersect Skew Lines: Two or more non-coplaner lines which never intersect

  13. A B C D E F H G Are points A , B , and G on the same plane? Yes

  14. Other examples of 3 intersecting planes:

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