1. Lesson 1-2 Point, Line, Plane 1 Lesson 1-2 Point, Line, Plane

2. Objectives/Assignment: Understand and use the basic undefined terms and defined terms of geometry. Sketch the intersections of lines and planes.

3. Using Undefined terms and definition A definition uses known words to describe a new word. In geometry, some words such as point, line and plane are undefined terms or not formally defined.

4. Using Undefined terms and definition A point has no dimension. It is usually represented by a small dot. A Point A

5. Lesson 1-1 Point, Line, Plane 5 Points Points do not have actual size. How to Sketch: Using dots How to label: Use capital letters Never name two points with the same letter (in the same sketch). A B AC

6. Lesson 1-1 Point, Line, Plane 6 Lines Lines extend indefinitely and have no thickness or width. How to sketch : using arrows at both ends. How to name: 2 ways (1) small script letter – line n (2) any two points on the line - Never name a line using three points - n A B C

7. Using Undefined terms and definition A plane extends in two dimensions. It is usually represented by a shape that looks like a tabletop or wall. You must imagine that the plane extends without end even though the drawing of a plane appears to have edges. A B C M Plane M or plane ABC

8. Lesson 1-1 Point, Line, Plane 8 Planes A plane is a flat surface that extends indefinitely in all directions. How to sketch: Use a parallelogram (four sided figure) How to name: 2 ways (1) Capital script letter – Plane M (2) Any 3 non collinear points in the plane - Plane: ABC/ ACB / BAC / BCA / CAB / CBA A B C Horizontal Plane M Vertical PlaneOther

9. A few basic concepts... Must be commonly understood without being defined. One such concept is the idea that a point lies on a line or a plane. Collinear points are points that lie on the same line. Coplanar points are points that lie on the same plane.

10. Ex. 1: Naming Collinear and Coplanar Points a. Name three points that are collinear Solution: D, E and F lie on the same line, so they are collinear. G D E F H

11. Ex. 1: Naming Collinear and Coplanar Points b. Name four points that are coplanar. Solution: D, E, F, and G lie on the same plane, so they are coplanar. Also D, E, F, and H are coplanar; although, the plane containing them is not drawn. G D E F H

12. Ex. 1: Naming Collinear and Coplanar Points c. Name three points that are not collinear. Solution: There are many correct answers. For instance, points H, E, and G do not lie on the same line. G D E F H

13. Lesson 1-1 Point, Line, Plane 13 Different planes in a figure: A B CD E F G H Plane ABCD Plane EFGH Plane BCGF Plane ADHE Plane ABFE Plane CDHG Etc.

14. Lesson 1-1 Point, Line, Plane 14 Other planes in the same figure: Any three non collinear points determine a plane! Plane AFGD Plane ACGE Plane ACH Plane AGF Plane BDG Etc.

15. Lesson 1-1 Point, Line, Plane 15 Coplanar Objects Coplanar objects (points, lines, etc.) are objects that lie on the same plane. The plane does not have to be visible. Are the following points coplanar? A, B, C ? A, B, C, F ? H, G, F, E ? E, H, C, B ? A, G, F ? C, B, F, H ? Yes No Yes No

16. More... Another undefined concept in geometry is the idea that a point on a line is between two other points on the line. You can use this idea to define other important terms in geometry. Consider the line AB (symbolized by AB). l Line l or AB

17. More... The line segment or segment AB (symbolized by AB) consists of the endpoints A and B, and all points on AB that are between A and B. l Line l or AB A A B B Segment AB

18. More... The ray AB (symbolized by AB) consists of the initial point A and all points on AB that lie on the same side of A as point B. l Line l or AB A A B B Ray AB

19. More... Note that AB is the same as BA and AB is the same as BA. However, AB and BA are not the same. They have different initial points and extend in different directions. l Line l or AB A A B B Ray BA

20. More... If C is between A and B, then CA and CB are opposite rays. Like points, segments and rays are collinear if they lie on the same line. So, any two opposite rays are collinear. Segments, rays and lines are coplanar if they lie on the same plane. l Line l or AB A C B

21. Ex. 2: Drawing lines, segments and rays Draw three noncollinear points J, K, and L. Then draw JK, KL and LJ. J K L Draw J, K and L Then draw JK

22. Ex. 2: Drawing lines, segments and rays Draw three noncollinear points J, K, and L. Then draw JK, KL and LJ. J K L Draw KL

23. Ex. 2: Drawing lines, segments and rays Draw three noncollinear points J, K, and L. Then draw JK, KL and LJ. J K L Draw LJ

24. Ex. 3: Drawing Opposite Rays Draw two lines. Label points on the lines and name two pairs of opposite rays. Solution: Points M, N, and X are collinear and X is between M and N. So XM and XN are opposite rays. P M Q N X

25. Ex. 3: Drawing Opposite Rays Draw two lines. Label points on the lines and name two pairs of opposite rays. Solution: Points P, Q, and X are collinear and X is between P and Q. So XP and XQ are opposite rays. P M Q N X

26. Goal 2: Sketching Intersections of Lines and Planes Two or more geometric intersect if they have one or more points in common. The intersection of the figures is the set of points the figures have in common. Activity: p. 12 – Modeling intersections. Use two index cards. Label them as shown and cut slots along each card. Complete the exercise and place the completed questions in your lab section labeling this Lab 1.2.

27. Ex. 4: Sketching intersections Sketch the figure described. a. A line that intersects a plane in one point  Draw a plane and a line.  Emphasize the point where they meet.  Dashes indicate where the line is hidden by the plane

28. Ex. 4: Sketching intersections Sketch the figure described. b. Two planes that intersect in a line  Draw two planes.  Emphasize the line where they meet.  Dashes indicate where one plane is hidden by the other plane.

29. Lesson 1-1 Point, Line, Plane 29 Intersection of Figures The intersection of two figures is the set of points that are common in both figures. The intersection of two lines is a point. m n P Continued……. Line m and line n intersect at point P.

30. Lesson 1-1 Point, Line, Plane 30 3 Possibilities of Intersection of a Line and a Plane (1) Line passes through plane – intersection is a point. (2) Line lies on the plane - intersection is a line. (3) Line is parallel to the plane - no common points.

31. Lesson 1-1 Point, Line, Plane 31 Intersection of Two Planes is a Line. P R A B Plane P and Plane R intersect at the line