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2.1 Points, Lines, and Planes
- 1. Points, Lines, & Planes Objectives • Identify and correctly label points, lines, line segments, rays, and planes.
- 2. pointpointpointpoint – a location in space, represented by a capital letter. (No dimension) • A Point A Notation: A point is always labeled with a capital printed letter. It has no size – one point is not bigger than another.another.
- 3. linelinelineline – a straight path that has no thickness and extends forever in both directions. (1 dimension) Example: • • A B m AB or BA or line m Notation: A line is labeled either by two points or a single lowercase letter. The order of the two letters does not matter. (AB is the same as BA.)
- 4. collinearcollinearcollinearcollinear – points that lie on the same line. Example: Points A, B, and C are collinear. • • A C B • D • noncollinearnoncollinearnoncollinearnoncollinear – points that do not lie on the same line. Points A, B, and D are noncollinear. Two points are always collinear. (note)
- 5. Examples 1. Are points G, H, and J collinear or noncollinear? 2. Are points F, H, and K collinear or F • H • G • J • K • 2. Are points F, H, and K collinear or noncollinear? 3. Are points J and K collinear or noncollinear?
- 6. Examples 1. Are points G, H, and J collinear or noncollinear? 2. Are points F, H, and K collinear or F • H • G • J • K • collinear 2. Are points F, H, and K collinear or noncollinear? 3. Are points J and K collinear or noncollinear? noncollinear collinear
- 7. line segment – part of a line consisting of two points, called endpointsendpointsendpointsendpoints, and all points between them. ray – part of a line that starts at an endpoint and extends forever in one direction. A B orAB BA Notation: The order of the letters does matter for the name of a ray. A B • AB AB is not the same as BA.
- 8. opposite rays – two rays that have a common endpoint and form a line. Q • R • S • and are opposite raysRS RQ You could also write RQ as QR (note the arrow direction), although it’s a little confusing to read.
- 9. planeplaneplaneplane – a flat surface that has no thickness and extends forever. (2 dimensions) A• B• C• R Plane ABC or plane R Notation: A plane is named either by a script capital letter or three noncollinear points.
- 10. coplanarcoplanarcoplanarcoplanar – points that lie in the same plane. Example: Points A, B, C, and D are coplanar. A • B • C • R E • D • Points A, B, C, and D are coplanar. noncoplanarnoncoplanarnoncoplanarnoncoplanar – points that do not lie in the same plane. Points A, B, C, and E are noncoplanar. Three noncollinear points are always coplanar.
- 11. Example Find three different ways you can name the plane below. S F R D E E T
- 12. Example Find three different ways you can name the plane below. S F R D E PlanePlanePlanePlane TTTT or any three letters exceptor any three letters exceptor any three letters exceptor any three letters except SSSS E T
- 13. undefined termundefined termundefined termundefined term – a basic figure that cannot be defined in terms of other figures Points, lines, and planes are undefined terms – all other geometric figures are defined in terms of them. Our “definitions” are just descriptions of their attributes. postulatepostulatepostulatepostulate – a statement that is accepted as true withoutpostulatepostulatepostulatepostulate – a statement that is accepted as true without proof. Also called an axiom.
- 14. Postulate: Through any two points there is exactly one line. (Euclid’s first postulate) Postulate: Through any three noncollinear points there is exactly one plane containing them. Postulate: If two lines intersect, then they intersect in exactly one point.exactly one point.
- 15. Postulate: If two planes intersect, then they intersect in exactly one line.
- 16. Examples: 1. The intersection of planes H and E is ____ or ____. 2. The intersection of m and n is ____. 3. Line k intersects E at ____. k E H m n R X S Y 4. R, X, and S are __________. n
- 17. Examples: 1. The intersection of planes H and E is ____ or ____. 2. The intersection of m and n is ____. 3. Line k intersects E at ____. n X Y k E H m n R X S Y RS 4. R, X, and S are __________.collinear n