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1. New Jersey Center for Teaching and Learning Progressive Mathematics Initiative This material is made freely available at www.njctl.org and is intended for the non-commercial use of students and teachers. These materials may not be used for any commercial purpose without the written permission of the owners. NJCTL maintains its website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning community, and/or provide access to course materials to parents, students and others. Click to go to website www.njctl.org

2. Points, Lines, Planes, & Angles October 4, 2011 www.njctl.org

3. Setting the PowerPoint View • Use Normal View for the Interactive Elements • To use the interactive elements in this presentation, do not select the Slide Show view. Instead, select Normal view and follow these steps to set the view as large as possible: • On the View menu, select Normal. • Close the Slides tab on the left. • In the upper right corner next to the Help button, click the ^ to minimize the ribbon at the top of the screen.  • On the View menu, confirm that Ruler is deselected. • On the View tab, click Fit to Window. • Use Slide Show View to Administer Assessment Items • To administer the numbered assessment items in this presentation, use the Slide Show view. (See Slide 17 for an example.)

4. Table of Contents Click on the topic to go to that section Points, Lines, & Planes Line Segments Pythagorean Theorem Distance between points Midpoint formula Locus & Constructions Angles & Angle Relationships Angle Addition Postulate Angle Bisectors & Constructions

5. Points, Lines, & Planes Return to Table of Contents

6. Definitions An "undefined term" is a word or term that does not require further explanation. There are three undefined terms in geometry: Points - A point has no dimensions (length, width, height), it is usually represent by a capital letter and a dot on a page. It shows position only. Lines - composed of an unlimited number of points along a straight path. A line has no width or height and extends infinitely in opposite directions. Planes - a flat surface that extends indefinitely in two-dimensions. A plane has no thickness.

7. Points & Lines A television picture is composed of many dots placed closely together. But if you look very closely, you will see the spaces. B A However, in geometry, a line is composed of an unlimited/infinite number of points. There are no spaces between the point that make a line.You can always find a point between any two other points. The line above would b called line or line

8. or Line , , all refer to the same line Line a Points are labeled with letters. (Points A, B, or C) Lines are named by using any two points OR by using a single lower-cased letter. Arrowheads show the line continues without end in opposite directions.

9. , , or Line , all refer to the same line Line a Collinear Points - Points D, E, and F above are called collinear points, meaning they all lie on the same line. Points A, B, and C are NOT collinear point since they do not lie on the same (one) line. Postulate: Any two points are always collinear.

10. Example Give six different names for the line that contains points U, V, and W. Answer (click)

11. Postulate: two lines intersect at exactly one point. If two non-parallel intersect they do so at only one point. and intersect at K.

12. Example a. Name three points that are collinear b. Name three sets of points that are noncollinear c. What is the intersection of the two lines? Answer a. A, D, C b. A,B,D / A,C,B / C,D,B (others)

13. Rays are also portions of a line. or is read ray AB. Rays start at an initial point, here endpoint A, and continues infinitely in one direction. Ray has a different initial point, endpoint B, and continues infinitely in the direction marked. Rays and are NOT the same. They have different initial points and extend in different directions.

14. Suppose point C is between points A and B Rays and are opposite rays. Opposite rays are two rays with a common endpoint that point in opposite directions and form a straight line. Recall: Since A, B, and C all lie on the same line, we know they are collinear points. Similarly, segments and rays are called collinear, if they lie on the same line. Segments, rays, and lines are also called coplanar if they all lie on the same plane.

15. Example Name a point that is collinear with the given points. a. R and P b. M and Q c. S and N d. O and P

16. Example Name two opposite rays on the given line e. f. g. h

17. 1 is the same as . True False Hint Read the notation carefully. Are they asking about lines, line segments, or rays?

18. 2 is the same as True False

19. 3 Line p contains just three points True False Answer Remember that even though only three points are marked, a line is composed of an infinite number of points. You can always find another point in between two other points.

20. 4 Points D, H, and E are collinear. True False

21. 5 Points G, D, and H are collinear. True False

22. 6 Ray LJ and ray JL are opposite rays. Yes No Explain your answer?

23. 7 Which of the following are opposite rays? A and B and C and D and

24. 8 Name the initial point of A J B K L C

25. 9 Name the initial point of A J B K L C

26. Are the three points collinear? If they are, name the line they lie on. a L, K, J b N, I, M c M, N, K d P, M, I

27. Planes Collinear points are points that are on the same line. F,G, and H are three collinear points. J,G, and K are three collinear points. J,G, and H are three non-collinear points. F, G, H, and I are coplanar. F, G, H, and J are also coplanar, but the plane is not drawn. Coplanar points are points that lie on the same plane. F,G, and H are coplanar in addition to being collinear. G, I, and K are non-coplanar and non-collinear. Any three noncollinear points can name a plane.

28. Planes can be named by any three noncollinear points: - plane KMN, plane LKM, or plane KNL - or, by a single letter such as Plane R (all name the same plane) Coplanar points are points that lie on the same plane: - Points K, M, and L are coplanar - Points O, K, and L are non-coplanar in the diagram However, you could draw a plane to contain any three points

29. Postulate: If two planes intersect, they intersect along exactly one line. A B The intersection of the two planes above is shown by line As another example, picture the intersections of the four walls in a room with the ceiling or the floor. You can imagine a line laying along the intersections of these planes.

30. Postulate: Through any three noncollinear points there is exactly one plane.

31. Example Name the following points: A point not in plane HIE A point not in plane GIE Two points in both planes Two points not on

32. 10 Line BC does not contain point R. Are points R, B, and C collinear? Yes No

33. 11 Plane LMN does not contain point P. Are points P, M, and N coplanar? Yes No Hint: what do we know about any three points?

34. 12 Plane QRS contains . Are points Q, R, S, and V coplanar? (Draw a picture) Yes No

35. 13 Plane JKL does not contain . Are points J, K, L, and N coplanar? Yes No

36. and intersect at 14 D A B C A C D B

37. 15 Which group of points are noncoplanar with points A,B, and F. E, F, B, A A B A, C, G, E D, H, G, C C D F, E, G, H

38. 16 Are lines and coplanar? Yes No Answer

39. Plane ABC and plane DCG intersect at ? 17 C A B line DC Line CG C D they don't intersect Answer

40. Planes ABC, GCD, and EGC intersect at ? 18 line A B point C point A C D line Answer

41. Name another point that is in the same plane as points E, G, and H 19 B A B C D C D F Answer

42. Name a point that is coplanar with points E, F, and C 20 H A B B D C D A Answer

43. 21 Intersecting lines are __________ coplanar. A Always B Sometimes Never C

44. 22 Two planes ____________ intersect at exactly one point. A Always B Sometimes Never C

45. 23 A plane can __________ be drawn so that any three points are coplaner A Always B Sometimes Never C

46. 24 A plane containing two points of a line __________ contains the entire line. A Always B Sometimes Never C

47. 25 Four points are ____________ noncoplanar. A Always B Sometimes Never C

48. 26 Two lines ________________ meet at more than one point. A Always Sometimes B C Never Look what happens if I place line y directly on top of line x. Hint

49. Line Segments Return to Table of Contents

50. Line Segments or Line segments are portions of a line. or endpoint endpoint is read segmentAB.. Line Segment or are different names for the same segment. It consists of the endpoints A and B and all the points on the line between them.