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Points

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. A. C. Lines. Lines extend indefinitely and have no thickness or width.

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Points

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  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 A C 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 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. A B C Collinear C A B Non collinear Lesson 1-1 Point, Line, Plane

  4. Postulate 1-5 Ruler 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 A B C B is the midpoint, so AB = BC

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

  7. Using the Segment Addition Postulate If DT = 60, find the value of x. Then find DS and ST. 2x - 8 3x - 12 D S T

  8. Using the Segment Addition Postulate If EG = 100, find the value of x. Then find EF and FG. 4x - 20 2x + 30 E F G

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

  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 Line m and line n intersect at point P. P n Continued……. 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 M B C Horizontal Plane Vertical Plane Other 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. 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). Lesson 1-1 Point, Line, Plane

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

  15. Different planes in a figure: A B Plane ABCD Plane EFGH Plane BCGF Plane ADHE Plane ABFE Plane CDHG Etc. D C E F H G 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. 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 ? Yes A, B, C, F ? No H, G, F, E ? Yes E, H, C, B ? Yes A, G, F ? Yes C, B, F, H ? No Lesson 1-1 Point, Line, Plane

  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

  19. Angle Relationships

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

  21. These are examples of adjacent angles. 45º 80º 35º 55º 130º 50º 85º 20º

  22. These angles are NOT adjacent. 100º 50º 35º 35º 55º 45º

  23. Complementary Anglessum to 90° 50° 40°

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

  25. Supplementary Anglessum to 180° 150° 30°

  26. Supplementary angles add up to 180º. 40º 120º 60º 140º Adjacent and Supplementary Angles Supplementary Anglesbut not Adjacent

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

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

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

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

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

  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. 1 2 3 4 6 5 7 8 Let’s Practice m<1=120° Find all the remaining angle measures. 120° 60° 120° 60° 120° 60° 120° 60°

  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. 2x° 3x°

  40. 3) Solve for x. 2x° 3x°

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

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

  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. 5x 4x

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

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

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

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