Points, Lines, and Planes KEY

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Ashlee Hodges
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1 ackground Historically, much of geometry was developed as uclidean geometry, or non-coordinate geometry. It was named after the Greek mathematician uclid. uclid s most important work was the 13 volumes of The lements of. He began his system of geometry with three undefined terms: point, line, and plane. From those terms he defined other geometric vocabulary and postulates. uclid then proceeded to prove theorems using the definitions and postulates, much as we do today. Geometric Vocabulary Undefined terms These terms can only be explained using examples and descriptions. These undefined terms can be used to define other geometric terms and properties. Term escription Naming Point Line Has no actual size, used to represent an object or location in space Has no thickness or width, used to represent a continuous set of linear points that extend indefinitely in both directions Named by a capital letter Named by a lowercase script letter or by two points on the line. Symbolic Representation m Plane Has no thickness, width, or depth, used to represent a flat surface that extends indefinitely in all directions. Named by a capital script letter or by three noncollinear points in the plane. Q Plane Q Plane 1. escribe three real world situations that depict the concept of a point. nswers will vary. 2. escribe three real world situations that characterize a line. nswers will vary. 3. escribe three real world situations that characterize a plane. nswers will vary. 2012, TS 05/15/12 page 1 of 6

2 efined terms ll other terms in geometry must be definable and a definition includes a category and then a list of critical attributes. xample: Space Set of all points, boundless and three-dimensional. Set of all points is the classification oundless and three dimensional are the critical attributes that make this definition different from other definitions. efined terms space Set of all points, boundless and three dimensional collinear Set of points, that all lie on the same line (Hint: Two points are always collinear. Three points must be checked to determine if they are collinear.) non-collinear Set of points, that do not all lie on the same line coplanar Set of points, or lines, that lie in the same plane (Hint: Three points are always coplanar. Four points must be checked to determine if they are coplanar.) non-coplanar Set of points, or lines, that do not lie in the same plane skew lines Two non-coplanar lines that do not intersect parallel lines Two coplanar lines that do not intersect Intersections of geometric terms Two lines intersect at a point Two planes intersect at a line line and a plane intersect at a point x V R P Q y 2012, TS 05/15/12 page 2 of 6

3 Guided Practice 1. How many lines can you draw through? Infinitely many 2. How many lines can you draw through and? One unique line, 3. raw and label, between and. How many different ways can you name line, ( )? Six (,,,,, ) 4. Give three ways to name the line that connects the points. F G H J nswers will vary. 5. How many different lines can you draw through three non-collinear points, when taken two at a time? Name them. Three lines can be drawn, lines,,, 6. List all possible names for the given figures. d a. R S T U b.,,,,,,,,,,, line d (also all lines associated with the given line segments) X Plane, Plane XYZ, Plane XZY, Plane YXZ, Plane YZX, Plane ZXY, Plane ZYX Z Y 2012, TS 05/15/12 page 3 of 6

4 F 7. Refer to the figure above to answer the questions. a. re,, and collinear? xplain. they are contained in at least two lines. b. re,,,, and coplanar? xplain. they are contained in at least two planes. c. How many planes appear in this figure? Name them. Five plane F or ;,,,, d. What is the intersection of and? Point e. What is the intersection of and plane F? Point f. What is the intersection of plane F and plane? Line 2012, TS 05/15/12 page 4 of 6

5 Practice Problems 1. re,, F, and G coplanar? 2. List four other coplanar points that lie in a different plane. nswers vary 3. re,, H, G, coplanar? 4. List four other non-coplanar points? nswers vary F Rectangular Prism H G 5. Name the plane that contains Q, R, S, and T in two different ways. Plane QRS, Plane RST, etc. 6. re N, P, Q, and S coplanar? O N 7. re M and R collinear? P 8. re P, N and Q collinear? M Rectangular 9. re LM and NM Prism coplanar? xplain., they can be contained in a flat surface. 10. re Q and R collinear? xplain., any two points are contained in one line. 11. re L, N, and R collinear? xplain., they cannot be contained in one line. 12. re LM M L and Q coplanar? xplain., they can be contained in a flat surface. 13. re NM Q and point R coplanar? xplain., they can be contained in a flat surface. 14. re NQ and NM coplanar? xplain., they can be contained in a flat surface. 15. What is the intersection of NM and LM? Point M 16. What is the intersection of NM and QN? Point N 17. What is the intersection of plane K and LM? Point M 18. What is the intersection of plane K and plane NMR? Line MN Q N R S T K R 2012, TS 05/15/12 page 5 of 6

6 Tell whether the lines are intersecting (I), Parallel (P) or Skew (S). 19. H and F 20. and H Parallel Intersecting 21. G and 22. G and Parallel Skew 23. Is plane parallel to plane G? 24. Is plane FG parallel to plane? 25. o planes FGH and intersect? 26. o planes G and FH intersect? F Rectangular Prism H G 2012, TS 05/15/12 page 6 of 6

1.1 Points, Lines, and Planes

1.1 Points, Lines, and Planes Identify and model points, lines and planes. Identify collinear, and coplanar points and intersecting lines and planes in the space. Vocabulary Point, line, plane, collinear,