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

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

3. Lesson 1-1 Point, Line, Plane 3 Collinear Points Collinear points are points that lie on the same line. (The line does not have to be visible.) A point lies on the line if the coordinates of the point satisfy the equation of the line. Ex: To find if A (1, 0) is collinear with the points on the line y = -3x + 3. Substitute x = 1 and y = 0 in the equation. 0 = -3 (1) + 3 0 = -3 + 3 0 = 0 The point A satisfies the equation, therefore the point is collinear with the points on the line. ABC A B C Collinear Non collinear

4. Postulate 1-5Ruler Postulate The distance between any two points is the absolute value of the difference of the corresponding numbers (on a number line or ruler) Congruent Segments: two segments with the same length

5. Midpoint: a point that divides the segment into two equal parts ABC B is the midpoint, so AB = BC

6. Postulate 1-6Segment Addition Postulate If three points A, B, and C are collinear and B is between A and C, then AB + BC = AC A B C

7. Using the Segment Addition Postulate If DT = 60, find the value of x. Then find DS and ST. DST 2x - 83x - 12

8. Using the Segment Addition Postulate If EG = 100, find the value of x. Then find EF and FG. EFG 4x - 202x + 30

9. Finding Lengths C is the midpoint of AB. Find AC, CB, and AB. A CB 2x + 1 3x – 4

10. Lesson 1-1 Point, Line, Plane 10 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.

11. Lesson 1-1 Point, Line, Plane 11 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

12. Lesson 1-1 Point, Line, Plane 12 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.

13. Lesson 1-1 Point, Line, Plane 13 Throughout New York City there are free movie nights in the park. The movie screen in the middle of the park is an example of a plane (2D) in space (3D).

14. Lesson 1-1 Point, Line, Plane 14 Intersection of Two Planes is a Line. P R A B Plane P and Plane R intersect at the line

15. Lesson 1-1 Point, Line, Plane 15 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.

16. Lesson 1-1 Point, Line, Plane 16 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.

17. Lesson 1-1 Point, Line, Plane 17 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

18. Regents Questions on Planes Go to jmap.org -> Resources by Topics- >Geometry->Planes Groups assigned G.G.1 through G.G.9 to complete and present whole class Lesson 1-1 Point, Line, Plane 18

19. Angle Relationships

20. Adjacent angles are “side by side” and share a common ray. 45º 15º

21. These are examples of adjacent angles. 55º 35º 50º130 º 80º 45º 85º 20º

22. These angles are NOT adjacent. 45º55º 50º 100 º 35º

23. Complementary Angles sum to 90° 40° 50°

24. Complementary angles add up to 90º. 60º 30º 40º 50º Adjacent and Complementary Angles Complementary Angles but not Adjacent

25. Supplementary Angles sum to 180° 30° 150 °

26. Supplementary angles add up to 180º. 60º120 º 40º 140 º Adjacent and Supplementary Angles Supplementary Angles but not Adjacent

27. Vertical Angles are opposite one another. Vertical angles are congruent. 100°

28. Vertical Angles are opposite one another. Vertical angles are congruent. 80°

29. Lines l and m are parallel. l || m 120 ° l m Note the 4 angles that measure 120°. n Line n is a transversal.

30. Lines l and m are parallel. l || m 60° l m Note the 4 angles that measure 60°. n Line n is a transversal.

31. Lines l and m are parallel. l || m 60° l m There are many pairs of angles that are supplementary. There are 4 pairs of angles that are vertical. 120 ° n Line n is a transversal.

32. If two lines are intersected by a transversal and any of the angle pairs shown below are congruent, then the lines are parallel. This fact is used in the construction of parallel lines.

33. Let’s Practice m<1=120° Find all the remaining angle measures. 1 4 2 6 5 78 3 60 ° 120 °

34. Practice Time!

35. 1) Find the missing angle. 36° ?°?°

36. 1) Find the missing angle. 36° ?°?°

37. 2) Find the missing angle. 64° ?°?°

38. 2) Find the missing angle. 64° ?°?°

39. 3) Solve for x. 3x° 2x°

40. 3) Solve for x. 3x° 2x°

41. 4) Solve for x. 2x + 5 x + 25

42. 4) Solve for x. 2x + 5 x + 25

43. 5) Find the missing angle. ?°?° 168°

44. 5) Find the missing angle. ?°?° 168°

45. 6) Find the missing angle. 58° ?°?°

46. 6) Find the missing angle. 58° ?°?°

47. 7) Solve for x. 4x 5x

48. 7) Solve for x. 4x 5x

49. 8) Solve for x. 2x + 10 3x + 20

50. 8) Solve for x. 2x + 10 3x + 20

51. 9) Lines l and m are parallel. l || m Find the missing angles. 42° l m b°b° d°d° f°f° a ° c°c° e°e° g°g°

52. 10) Lines l and m are parallel. l || m Find the missing angles. 81° l m b°b° d°d° f°f° a ° c°c° e°e° g°g°

53. In the figure a || b. 13. Name the angles congruent to  3. 14. Name all the angles supplementary to  6. 15. If m  1 = 105° what is m  3? 16. If m  5 = 120° what is m  2?

54. Find the missing angles. 70 ° b° 70 ° d °65 ° Hint: The 3 angles in a triangle sum to 180°.

55. Find the missing angles. 70 ° 40° 70 ° 75 ° 65 ° Hint: The 3 angles in a triangle sum to 180°.

56. Find the missing angles. 45 ° b° 50 ° d °75 ° Hint: The 3 angles in a triangle sum to 180°.

57. Find the missing angles. 45 ° 85° 50 ° 20°75 ° Hint: The 3 angles in a triangle sum to 180°.

58. Another practice problem Find all the missing angle measures, and name the postulate or theorem that gives us permission to make our statements. 40 ° 120°

59. Ex. : Interior Angles of a Quadrilateral 80° 70° 2x° x°x° Find the measure of each angle in the quadrilateral?

60. Find the measure of the missing angle in the figure below 100  135  70  x 135 + 100 + 70 + x = quadrilateral 360 305 + x = 360 -305 x = 55 

61. The End