1. Eleanor Roosevelt High School Chin-Sung Lin

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

3. Mr. Chin-Sung Lin ERHS Math Geometry

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

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

6. There is exactly one plane containing a line and a point not on the line Mr. Chin-Sung Lin ERHS Math Geometry m A B P

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

8. 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 Mr. Chin-Sung Lin ERHS Math Geometry m n

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

10. 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 intercepting lines, and skew lines in the cube in the cube Mr. Chin-Sung Lin ERHS Math Geometry A B D C E F H G

11. Mr. Chin-Sung Lin ERHS Math Geometry

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

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

14. 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 Mr. Chin-Sung Lin ERHS Math Geometry C A B D

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

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

17. 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, k  n k  m, and k  n, k  s then k  s Mr. Chin-Sung Lin ERHS Math Geometry n p k s m

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

19. 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 Mr. Chin-Sung Lin ERHS Math Geometry A P k m B

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

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

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

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

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

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

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

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

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

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

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

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

32. 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 Mr. Chin-Sung Lin ERHS Math Geometry A B D p q C

33. 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 Mr. Chin-Sung Lin ERHS Math Geometry A B D p q C

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

35. 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 Mr. Chin-Sung Lin ERHS Math Geometry p A B

36. Mr. Chin-Sung Lin ERHS Math Geometry p A B C D 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

37. 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 Mr. Chin-Sung Lin ERHS Math Geometry P A B

38. 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 Mr. Chin-Sung Lin ERHS Math Geometry A B R P n Q m

39. 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 Mr. Chin-Sung Lin ERHS Math Geometry p A B C D q

40. 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 Mr. Chin-Sung Lin ERHS Math Geometry p A B C q

41. 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 Mr. Chin-Sung Lin ERHS Math Geometry p A B C D q E

42. Mr. Chin-Sung Lin ERHS Math Geometry

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

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

45. If a plane intersects two parallel planes, then the intersection is two parallel lines Mr. Chin-Sung Lin ERHS Math Geometry n m p

46. 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 Mr. Chin-Sung Lin ERHS Math Geometry n m A B C D p

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

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

49. 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 Mr. Chin-Sung Lin ERHS Math Geometry p q L M A B

50. 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 Mr. Chin-Sung Lin ERHS Math Geometry q B p A

51. 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 Mr. Chin-Sung Lin ERHS Math Geometry q p A B R s

52. If two planes are parallel, then a line perpendicular to one of the planes is perpendicular to the other Given: Plane p parallel to plane q, and AB⊥p and intersecting plane q at B Prove: q⊥AB Mr. Chin-Sung Lin ERHS Math Geometry q B p A

53. If two planes are parallel, then a line perpendicular to one of the planes is perpendicular to the other Given: Plane p parallel to plane q, and AB⊥p and intersecting plane q at B Prove: q⊥AB Mr. Chin-Sung Lin ERHS Math Geometry B A E C q p

54. If two planes are parallel, then a line perpendicular to one of the planes is perpendicular to the other Given: Plane p parallel to plane q, and AB⊥p and intersecting plane q at B Prove: q⊥AB Mr. Chin-Sung Lin ERHS Math Geometry B A E C q p F D

55. Two planes are perpendicular to the same line if and only if the planes are parallel Mr. Chin-Sung Lin ERHS Math Geometry q B p A

56. The distance between two planes is the length of the line segment perpendicular to both planes with an endpoint on each plane Mr. Chin-Sung Lin ERHS Math Geometry B A q p

57. Parallel planes are everywhere equidistant Given: Parallel planes p and q, with AC and BD each perpendicular to p and q with an endpoint on each plane Prove: AC = BD Mr. Chin-Sung Lin ERHS Math Geometry C A B q p D

58. Mr. Chin-Sung Lin ERHS Math Geometry

59. A polyhedron is a three-dimensional figure formed by the union of the surfaces enclosed by plane figures A polyhedron is a figure that is the union of polygons Mr. Chin-Sung Lin ERHS Math Geometry

60. Faces: the portions of the planes enclosed by a plane figure Edges: The intersections of the faces Vertices: the intersections of the edges Mr. Chin-Sung Lin ERHS Math Geometry Vertex Edge Face

61. A prism is a polyhedron in which two of the faces, called the bases of the prism, are congruent polygons in parallel planes Mr. Chin-Sung Lin ERHS Math Geometry

62. Lateral sides: the surfaces between corresponding sides of the bases Lateral edges: the common edges of the lateral sides Altitude: a line segment perpendicular to each of the bases with an endpoint on each base Height: the length of an altitude Mr. Chin-Sung Lin ERHS Math Geometry Lateral Side Lateral Edge Altitude/Height Base

63. The lateral edges of a prism are congruent and parallel Mr. Chin-Sung Lin ERHS Math Geometry Lateral Edges

64. A right prism is a prism in which the lateral sides are all perpendicular to the bases All of the lateral sides of a right prism are rectangles Mr. Chin-Sung Lin ERHS Math Geometry Lateral Sides

65. A parallelepiped is a prism that has parallelograms as bases Mr. Chin-Sung Lin ERHS Math Geometry

66. A rectangular parallelepiped is a parallelepiped that has rectangular bases and lateral edges perpendicular to the bases Mr. Chin-Sung Lin ERHS Math Geometry

67. A rectangular parallelepiped is also called a rectangular solid, and it is the union of six rectangles. Any two parallel rectangles of a rectangular solid can be the bases Mr. Chin-Sung Lin ERHS Math Geometry

68. The lateral area of the prism is the sum of the areas of the lateral faces The total surface area is the sum of the lateral area and the areas of the bases Mr. Chin-Sung Lin ERHS Math Geometry

69. Calculate the lateral area of the prism Calculate the total surface area of the prism Mr. Chin-Sung Lin ERHS Math Geometry 4 7 5

70. Area of the bases:7 x 5 x 2 = 70 Lateral area:2 x (4 x 5 + 4 x 7) = 96 Total surface area:70 + 96 = 166 Mr. Chin-Sung Lin ERHS Math Geometry 4 7 5

71. The bases of a right prism are equilateral triangles Calculate the lateral area of the prism Calculate the total surface area of the prism Mr. Chin-Sung Lin ERHS Math Geometry 5 4

72. Area of the bases:½ x (4 x 2√3) x 2= 8√3 Lateral area:3 x (4 x 5) = 60 Total surface area:60 + 8√3 ≈ 73.86 Mr. Chin-Sung Lin ERHS Math Geometry 5 4 2 2√3 4

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74. The volume (V) of a prism is equal to the area of the base (B) times the height (h) V = B x h Mr. Chin-Sung Lin ERHS Math Geometry Base (B) Height (h)

75. A right prism is shown in the diagram Calculate the Volume of the prism Mr. Chin-Sung Lin ERHS Math Geometry 5 4 2

76. A right prism is shown in the diagram Calculate the Volume of the prism B = ½ x 4 x 2 = 4 h = 5 V = Bh = 4 x 5 = 20 Mr. Chin-Sung Lin ERHS Math Geometry 5 4 2

77. A right prism is shown in the diagram Calculate the Volume of the prism Mr. Chin-Sung Lin ERHS Math Geometry 3 5 4

78. A right prism is shown in the diagram Calculate the Volume of the prism B = 5 x 4 = 20 h = 3 V = Bh = 20 x 3 = 60 Mr. Chin-Sung Lin ERHS Math Geometry 3 5 4

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80. A pyramid is a solid figure with a base that is a polygon and lateral faces that are triangles Mr. Chin-Sung Lin ERHS Math Geometry

81. Vertex: All lateral edges meet in a point Altitude: the perpendicular line segment from the vertex to thebase Mr. Chin-Sung Lin ERHS Math Geometry Vertex Altitude Vertex Altitude

82. A pyramid whose is a regular polygon and whose altitude is perpendicular to the base at its center A pyramid whose base is a regular polygon and whose altitude is perpendicular to the base at its center The lateral edges of a regular polygon are congruent The lateral faces of a regular pyramid are isosceles triangles The length of the altitude of a triangular lateral face is the slant height of the pyramid Mr. Chin-Sung Lin ERHS Math Geometry Slant Height Altitud e

83. The lateral area of a pyramid is the sum of the areas of the faces (isosceles triangles) The total surface area is the lateral area plus the area of the base Mr. Chin-Sung Lin ERHS Math Geometry Slant Height

84. The volume (V) of a pyramid is equal to of the area of the base (B) times the height (h) The volume (V) of a pyramid is equal to one third of the area of the base (B) times the height (h) V = (1/3) x B x h Mr. Chin-Sung Lin ERHS Math Geometry Base Area Height

85. A regular pyramid has a square base. The length of an edge of the base is 10 centimeters and the length of the altitude to the base of each lateral side is 13 centimeters a.What is the total surface area of the pyramid? b.What is the volume of the pyramid? Mr. Chin-Sung Lin ERHS Math Geometry 13 10

86. A regular pyramid has a square base. The length of an edge of the base is 10 centimeters and the length of the altitude to the base of each lateral side is 13 centimeters a.What is the total surface area of the pyramid? b.What is the volume of the pyramid? Mr. Chin-Sung Lin ERHS Math Geometry 13 10

87. A regular pyramid has a square base. The length of an edge of the base is 10 centimeters and the length of the altitude to the base of each lateral side is 13 centimeters a.What is the total surface area of the pyramid? b.What is the volume of the pyramid? Mr. Chin-Sung Lin ERHS Math Geometry 13 10 5 12

88. a. Total surface area: Lateral Area: ½ x 10 x 13 x 4 = 260 Base Area: 10 x 10 = 100 Total Area = 260 + 100 = 360 cm 2 b. Volume: B = 100 h = 12 V = (1/3) x 100 x 12 = 400 cm 3 Mr. Chin-Sung Lin ERHS Math Geometry 13 10 5 12

89. The base of a regular pyramid is a regular polygon and the altitude is perpendicular to the base at its center The center of a regular polygon is defined as the point that is equidistant to its vertices The lateral faces of a regular pyramid are isosceles triangles The lateral faces of a regular pyramid are congruent Mr. Chin-Sung Lin ERHS Math Geometry

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91. The solid figure formed by the congruent parallel curves and the surface that joins them is called a cylinder Mr. Chin-Sung Lin ERHS Math Geometry

92. Bases: the closed curves Lateral surface: the surface that joins the bases Altitude: a line segment perpendicular to the bases with endpoints on the bases Height: the length of an altitude Mr. Chin-Sung Lin ERHS Math Geometry Bases Lateral Surface Altitude

93. A cylinder whose bases are A cylinder whose bases are congruent circles Mr. Chin-Sung Lin ERHS Math Geometry

94. If the line segment joining the centers of the circular bases is perpendicular to the bases, the cylinder is a If the line segment joining the centers of the circular bases is perpendicular to the bases, the cylinder is a right circular cylinder Mr. Chin-Sung Lin ERHS Math Geometry

95. Base Area: 2πr 2 Lateral Area: 2πrh Total Surface Area: 2πrh + 2πr 2 Mr. Chin-Sung Lin ERHS Math Geometry r h

96. Volume: B x h = πr 2 h Mr. Chin-Sung Lin ERHS Math Geometry

97. A right cylinder as shown in the diagram. Calculate the total Surface Area Calculate the volume Mr. Chin-Sung Lin ERHS Math Geometry 6 14

98. Base Area: 2πr 2 = 2π6 2 ≈ 226.19 Lateral Area: 2πrh = 2π (6)(14) ≈ 527.79 Total Surface Area: 226.19 + 527.79 = 754.58 Volume: B x h = πr 2 h = π(6 2) (14) = 1583.36 Mr. Chin-Sung Lin ERHS Math Geometry 6 14

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100. Line OQ is perpendicular to plane p at O, and a point P is on plane p Keeping point Q fixed, move P through a circle on p with center at O. The surface generated by PQ is a right circular conical surface * A conical surface extends infinitely Mr. Chin-Sung Lin ERHS Math Geometry A C P O Q p

101. The part of the conical surface generated by PQ from plane p to Q is called a right circular cone Q:vertex of the cone Circle O: base of the cone OQ: altitude of the cone OQ: height of the cone, and PQ: slant height of the cone Mr. Chin-Sung Lin ERHS Math Geometry A A C P O Q p

102. Base Area: B = πr 2 Lateral Area: L = ½ Ch s = ½ (2πr)h s = πrh s Total Surface Area: πrh s + πr 2 * h s :slant height * h c :height * r:radius * B:base area * C:circumference Mr. Chin-Sung Lin ERHS Math Geometry A A C hshs C r p hchc B

103. Base Area: B = πr 2 Volume: V = ⅓ Bh c = ⅓ πr 2 h c * h s :slant height * h c :height * r:radius * B:base area * C:circumference Mr. Chin-Sung Lin ERHS Math Geometry A A C hshs B r p hchc C

104. Calculate the base area, lateral area, and total area Mr. Chin-Sung Lin ERHS Math Geometry A A C 26 10 p 24

105. Calculate the base area, lateral area, and total area Base Area: B = π(10) 2 = 100π Lateral Area: L = π(10)(26) = 260π Total Surface Area: 100π + 260π = 360π Mr. Chin-Sung Lin ERHS Math Geometry A A C 26 10 p 24

106. A cone and a cylinder have equal volumes and equal heights. If the radius of the base of the cone is 3 centimeters, what is the radius of the base of the cylinder? Volume of Cylinder: V = h = πr 2 h Volume of Cone: V = ⅓ π3 2 h = 3πh πr 2 h = 3πh, r 2 = 3, r = √3 cm Mr. Chin-Sung Lin ERHS Math Geometry A A C 3 cm p h r h

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108. A sphere is the set of all points equidistant from a fixed point called the center The radius of a sphere is the length of the line segment from the center of the sphere to any point on the sphere Mr. Chin-Sung Lin ERHS Math Geometry r O

109. If the distance of a plane from the center of a sphere is d and the radius of the sphere is r Mr. Chin-Sung Lin ERHS Math Geometry P O p d r P O p d r P O p d r r < d no points in common r = d one points in common r > d infinite points in common (circle)

110. A circle is the set of all points in a plane equidistant from a fixed point in the plane called the center Mr. Chin-Sung Lin ERHS Math Geometry O p r

111. The intersection of a sphere and a plane through the center of the sphere is a circle whose radius is equal to the radius of the sphere Mr. Chin-Sung Lin ERHS Math Geometry O p r r

112. A great circle of a sphere is the intersection of a sphere and a plane through the center of the sphere Mr. Chin-Sung Lin ERHS Math Geometry O p r r

113. If the intersection of a sphere and a plane does not contain the center of the sphere, then the intersection is a circle Given: A sphere with center at O plane p intersecting plane p intersecting the sphere at A and B the sphere at A and B Prove: The intersection is a circle Mr. Chin-Sung Lin ERHS Math Geometry O p C A B

114. If the intersection of a sphere and a plane does not contain the center of the sphere, then the intersection is a circle Given: A sphere with center at O plane p intersecting plane p intersecting the sphere at A and B the sphere at A and B Prove: The intersection is a circle Mr. Chin-Sung Lin ERHS Math Geometry O p r C A B

115. Mr. Chin-Sung Lin StatementsReasons 1. Draw a line OC, point C on plane p 1. Given, create two triangles OC  AC, OC  BC 2.  OCA and  OCB are right angles2. Definition of perpendicular 3. OA  OB3. Radius of a sphere 4. OC  OC4. Reflexive postulate 5.  OAC   OBC5. HL postulate 6. CA  CB 6. CPCTC 7. The intersection is a circle7. Definition of circles ERHS Math Geometry O p r C A B

116. The intersection of a plane and a sphere is a circle A great circle is the largest circle that can be drawn on a sphere Mr. Chin-Sung Lin ERHS Math Geometry O p p’

117. If two planes are equidistant from the center of a sphere and intersect the sphere, then the intersections are congruent circles Mr. Chin-Sung Lin ERHS Math Geometry O q p A B C D

118. Surface Area: S = 4πr 2 r:radius r:radius Mr. Chin-Sung Lin ERHS Math Geometry A r O

119. Volume: V = 4 / 3 πr 3 r:radius r:radius Mr. Chin-Sung Lin ERHS Math Geometry A r O

120. Find the surface area and the volume of a sphere whose radius is 6 cm Mr. Chin-Sung Lin ERHS Math Geometry A r O

121. Find the surface area and the volume of a sphere whose radius is 6 cm Surface Area: S = 4π6 2 = 144π cm 2 Volume: V = 4 / 3 π6 3 = 288π cm 3 Mr. Chin-Sung Lin ERHS Math Geometry A r O

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