1 / 123

The Geometry of Three Dimensions

Geometry Chap 11. The Geometry of Three Dimensions. Eleanor Roosevelt High School Chin-Sung Lin. ERHS Math Geometry. The Geometry of Three Dimensions. The geometry of three dimensions is called solid geometry. Mr. Chin-Sung Lin. ERHS Math Geometry. Points, Lines, and Planes.

carver
Download Presentation

The Geometry of Three Dimensions

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Geometry Chap11 The Geometry of Three Dimensions Eleanor Roosevelt High School Chin-Sung Lin

  2. ERHS Math Geometry The Geometry of Three Dimensions • The geometry of three dimensions is called • solid geometry Mr. Chin-Sung Lin

  3. ERHS Math Geometry Points, Lines, and Planes Mr. Chin-Sung Lin

  4. ERHS Math Geometry Postulates of the Solid Geometry • There is one and only one plane containing three non-collinear points B A C Mr. Chin-Sung Lin

  5. ERHS Math Geometry Postulates of the Solid Geometry • A plane containing any two points contains all of the points on the line determined by those two points B A m Mr. Chin-Sung Lin

  6. ERHS Math Geometry Theorems of the Points, Lines & Planes • There is exactly one plane containing a line and a point not on the line B m A P Mr. Chin-Sung Lin

  7. ERHS Math Geometry Theorems of the Points, Lines & Planes • If two lines intersect, then there is exactly one plane containing them • Two intersecting lines determine a plane m P A B n Mr. Chin-Sung Lin

  8. ERHS Math Geometry Parallel Lines in Space • Lines in the same plane that have no points in common • Two lines are parallel if and only if they are coplanar and have no points in common m n Mr. Chin-Sung Lin

  9. ERHS Math Geometry Skew Lines in Space • Skew lines are lines in space that are neither parallel nor intersecting n m Mr. Chin-Sung Lin

  10. ERHS Math Geometry Example • Both intersecting lines and parallel lines lie in a plane • Skew lines do not lie in a plane • Identify the parallel lines, • intercepting lines, and skew lines • in the cube H G D C E F A B Mr. Chin-Sung Lin

  11. ERHS Math Geometry Perpendicular Lines and Planes Mr. Chin-Sung Lin

  12. ERHS Math Geometry Postulates of the Solid Geometry • If two planes intersect, then they intersect in exactly one line B A Mr. Chin-Sung Lin

  13. ERHS Math Geometry Dihedral Angle • A dihedral angle is the union of two half-planes with a common edge Mr. Chin-Sung Lin

  14. ERHS Math Geometry • The measure of the plane angle formed by two rays each in a different half-plane of the angle and each perpendicular to the common edge at the same point of the edge • AC  AB and AD  AB • The measure of the dihedral angle: • mCAD The Measure of a Dihedral Angle C B D A Mr. Chin-Sung Lin

  15. ERHS Math Geometry • Perpendicular planes are two planes that intersect to form a right dihedral angle • AC  AB, AD  AB, and • AC  AD (mCAD = 90) • then • m  n Perpendicular Planes C m B n D A Mr. Chin-Sung Lin

  16. ERHS Math Geometry Theorems of Perpendicular Lines & Planes • If a line not in a plane intersects the plane, then it intersects in exactly one point k P A B n Mr. Chin-Sung Lin

  17. ERHS Math Geometry • A line is perpendicular to a plane if and only if it is perpendicular to each line in the plane through the intersection of the line and the plane • A plane is perpendicular to a line if the line is perpendicular to the plane • k  m, and k  n, • then k  s A Line is Perpendicular to a Plane k n s m p Mr. Chin-Sung Lin

  18. ERHS Math Geometry Postulates of the Solid Geometry • At a given point on a line, there are infinitely many lines perpendicular to the given line q p k r n m A Mr. Chin-Sung Lin

  19. ERHS Math Geometry Theorems of Perpendicular Lines & Planes • If a line is perpendicular to each of two intersecting lines at their point of intersection, then the line is perpendicular to the plane determined by these lines k m A P B Mr. Chin-Sung Lin

  20. ERHS Math Geometry Theorems of Perpendicular Lines & Planes Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k  AP and k  BP Prove: k  m k m A P B Mr. Chin-Sung Lin

  21. ERHS Math Geometry Theorems of Perpendicular Lines & Planes • Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k  AP and k  BP • Prove: k  m • Connect AB • Connect PT and • intersects AB at Q • Make PR = PS k R m A Q T P B S Mr. Chin-Sung Lin

  22. ERHS Math Geometry Theorems of Perpendicular Lines & Planes • Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k  AP and k  BP • Prove: k  m • Connect RA, SA • SAS • ΔRAP = ΔSAP k R m A Q T P B S Mr. Chin-Sung Lin

  23. ERHS Math Geometry Theorems of Perpendicular Lines & Planes Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k  AP and k  BP Prove: k  m CPCTC AR = AS k R m A Q T P B S Mr. Chin-Sung Lin

  24. ERHS Math Geometry Theorems of Perpendicular Lines & Planes • Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k  AP and k  BP • Prove: k  m • Connect RB, SB • SAS • ΔRBP = ΔSBP k R m A Q T P B S Mr. Chin-Sung Lin

  25. ERHS Math Geometry Theorems of Perpendicular Lines & Planes Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k  AP and k  BP Prove: k  m CPCTC BR = BS k R m A Q T P B S Mr. Chin-Sung Lin

  26. ERHS Math Geometry Theorems of Perpendicular Lines & Planes • Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k  AP and k  BP • Prove: k  m • SSS • ΔRAB = ΔSAB k R m A Q T P B S Mr. Chin-Sung Lin

  27. ERHS Math Geometry Theorems of Perpendicular Lines & Planes Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k  AP and k  BP Prove: k  m CPCTC RAB = SAB k R m A Q T P B S Mr. Chin-Sung Lin

  28. ERHS Math Geometry Theorems of Perpendicular Lines & Planes • Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k  AP and k  BP • Prove: k  m • Connect RQ, SQ • SAS • ΔRAQ = ΔSAQ k R m A Q T P B S Mr. Chin-Sung Lin

  29. ERHS Math Geometry Theorems of Perpendicular Lines & Planes • Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k  AP and k  BP • Prove: k  m • CPCTC • QR = QS k R m A Q T P B S Mr. Chin-Sung Lin

  30. ERHS Math Geometry Theorems of Perpendicular Lines & Planes • Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k  AP and k  BP • Prove: k  m • SSS • ΔRPQ = ΔSPQ k R m A Q T P B S Mr. Chin-Sung Lin

  31. ERHS Math Geometry Theorems of Perpendicular Lines & Planes • Given: A plane m determined by AP and BP, two lines that intersect at P. Line k such that k  AP and k  BP • Prove: k  m • CPCTC • mRPQ = mSPQ • mRPQ + mSPQ = 180 • mRPQ = mSPQ = 90 k R m A Q T B P S Mr. Chin-Sung Lin

  32. ERHS Math Geometry Theorems of Perpendicular Lines & Planes If two planes are perpendicular to each other, one plane contains a line perpendicular to the other plane Given: Plane p  plane q Prove: A line in p is perpendicular to q and a line in q is perpendicular to p C p B A q D Mr. Chin-Sung Lin

  33. ERHS Math Geometry Theorems of Perpendicular Lines & Planes If a plane contains a line perpendicular to another plane, then the planes are perpendicular Given: AC in plane p and AC  q Prove: p  q C p B A q D Mr. Chin-Sung Lin

  34. ERHS Math Geometry Theorems of Perpendicular Lines & Planes Two planes are perpendicular if and only if one plane contains a line perpendicular to the other C p B A q D Mr. Chin-Sung Lin

  35. ERHS Math Geometry Theorems of Perpendicular Lines & Planes Through a given point on a plane, there is only one line perpendicular to the given plane Given: Plane p and AB  p at A Prove: AB is the only line perpendicular to p at A B A p Mr. Chin-Sung Lin

  36. ERHS Math Geometry Theorems of Perpendicular Lines & Planes Through a given point on a plane, there is only one line perpendicular to the given plane Given: Plane p and AB  p at A Prove: AB is the only line perpendicular to p at A q B C A D p Mr. Chin-Sung Lin

  37. ERHS Math Geometry Theorems of Perpendicular Lines & Planes Through a given point on a line, there can be only one plane perpendicular to the given line Given: Any point P on AB Prove: There is only one plane perpendicular to AB A P B Mr. Chin-Sung Lin

  38. ERHS Math Geometry Theorems of Perpendicular Lines & Planes Through a given point on a line, there can be only one plane perpendicular to the given line Given: Any point P on AB Prove: There is only one plane perpendicular to AB A Q m n P R B Mr. Chin-Sung Lin

  39. ERHS Math Geometry Theorems of Perpendicular Lines & Planes If a line is perpendicular to a plane, then any line perpendicular to the given line at its point of intersection with the given plane is in the plane Given: AB  p at A and AB  AC Prove: AC is in plane p q B C A D p Mr. Chin-Sung Lin

  40. ERHS Math Geometry Theorems of Perpendicular Lines & Planes If a line is perpendicular to a plane, then every plane containing the line is perpendicular to the given plane Given: Plane p with AB  p at A, and C any point not on p Prove: Plane q determined by A, B, and C is perpendicular to p q B C A p Mr. Chin-Sung Lin

  41. ERHS Math Geometry Theorems of Perpendicular Lines & Planes If a line is perpendicular to a plane, then every plane containing the line is perpendicular to the given plane Given: Plane p with AB  p at A, and C any point not on p Prove: Plane q determined by A, B, and C is perpendicular to p q B C A D p E Mr. Chin-Sung Lin

  42. ERHS Math Geometry Parallel Lines and Planes Mr. Chin-Sung Lin

  43. ERHS Math Geometry Parallel Planes • Parallel planes are planes that have no points in common m n Mr. Chin-Sung Lin

  44. ERHS Math Geometry A Line is Parallel to a Plane • A line is parallel to a plane if it has no points in common with the plane k m Mr. Chin-Sung Lin

  45. ERHS Math Geometry Theorems of Parallel Lines & Planes • If a plane intersects two parallel planes, then the intersection is two parallel lines p m n Mr. Chin-Sung Lin

  46. ERHS Math Geometry Theorems of Parallel Lines & Planes • If a plane intersects two parallel planes, then the intersection is two parallel lines • Given: Plane p intersects plane m at AB • and plane n at CD, m//n • Prove: AB//CD p B A m D C n Mr. Chin-Sung Lin

  47. ERHS Math Geometry Theorems of Parallel Lines & Planes • Two lines perpendicular to the same plane are parallel • Given: Plane p, LA⊥p at A, and MB⊥p at B • Prove: LA//MB q M L B A p Mr. Chin-Sung Lin

  48. ERHS Math Geometry Theorems of Parallel Lines & Planes • Two lines perpendicular to the same plane are parallel • Given: Plane p, LA⊥p at A, and MB⊥p at B • Prove: LA//MB q N M L B A D C p Mr. Chin-Sung Lin

  49. ERHS Math Geometry Theorems of Parallel Lines & Planes • Two lines perpendicular to the same plane are coplanar • Given: Plane p, LA⊥p at A, and MB⊥p at B • Prove: LA and MB are coplanar q M L B A p Mr. Chin-Sung Lin

  50. ERHS Math Geometry Theorems of Parallel Lines & Planes • If two planes are perpendicular to the same line, then they are parallel • Given: Plane p⊥AB at A and q⊥AB at B • Prove: p//q A p B q Mr. Chin-Sung Lin

More Related