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Theorems

1.0 If a line is perpendicular to each of two intersecting lines at their point of intersection, it is perpendicular to the plane of the two lines.

1.1 Through a given point in a given line there can be one plane, and only one, perpendicular to the plane.

1.2 Through a given external point there can be one plane, and only one, perpendicular to a given line.

1.3 All the perpendiculars to a line at a point in the line lie in a plane which is perpendicular to the line at the point, and only one.

2.0 Through a given point in a plane one perpendicular, and only one, can be drawn to the plane.

3.0 Through a given external point one perpendicular, and only one, can be drawn to a plane.

4.0 If lines are drawn from a point in a perpendicular to a plane and meet the plane at equal distances from the foot of the perpendicular, they are equal.

5.0 If two lines drawn from a point in a perpendicular to a plane meet the plane at unequal distances from the foot of the perpendicular, the more remote is the greater.

5.1 The perpendicular is the shortest line segment from a point to a plane.

6.0 The locus of points equidistant from two given points is the plane perpendicular to the line joining them, at its midpoint.

7.0 The locus of points equidistant from the vertices of a triangle is the line through the center of the circumcircle, perpendicular to the plane of the triangle.

8.0 Two lines perpendicular to the same plane are parallel.

8.1 If one of two parallel lines is perpendicular to a plane, the other is also perpendicular to the plane.

8.2 If two lines are parallel to a third line, they are parallel to each other.

9.0 If two lines are parallel, every plane containing one of the lines, and only one, is parallel to the other line.

9.1 If a line is parallel to a plane, it is parallel to the intersection of that plane with any plane containing the line.

9.2 If a line and a plane are parallel, a parallel to the line through any point in the plane lies in the plane.

9.3 Through either of two skew lines there can be one plane, and only one, parallel to the other line.

9.4 Through a given point in space there can be one plane, and only one, parallel to two skew lines, or else parallel to one line and containing the other.

10.0 If two parallel planes are cut by a third plane, the lines of intersection are parallel.

11.0 Two planes perpendicular to the same line are parallel.

12.0 If a line is perpendicular to one of two parallel planes, it is perpendicular to the other.

12.1 Through a point outside a plane there can be one plane, and only one, parallel to a given plane.

12.2 Two parallel planes are everywhere equidistant.

12.3 If two planes are perpendicular to two intersecting lines, they will intersect.

13.0 If two intersecting lines are each parallel to a plane, the plane of these lines is parallel to that plane.

14.0 If two angles not in the same plane have their sides parallel and extending in the same direction, they are equal and their planes are parallel.

15.0 If two lines are cut by three parallel planes, their corresponding segments are proportional.

16.0 Two dihedral angles are equal if their plane angles are equal.

17.0 If a line is perpendicular to a given plane, every plane which contains this line is perpendicular to the given plane.

18.0 If two planes are perpendicular to each other, a line drawn in one of them perpendicular to their intersection is perpendicular to the other plane.

18.1 If two planes are perpendicular to each other, a line perpendicular to one of them at any point of their intersection will lie in the other.

18.2 If Two planes are perpendicular to each other, a line drawn perpendicular to one of them through any point of the other will lie in this other plane.

19.0 If two intersecting planes are perpendicular to a third plane, their intersection is also perpendicular to that plane.

19.1 Through a line not perpendicular to a plane there can be drawn one plane, and only one, perpendicular to a given plane.

20.0 The projection on a plane of a straight line not perpendicular to the plane is the line of intersection of the given plane with a plane that is perpendicular to it and containing the given line.

21.0 The locus of points equidistant from the faces of a dihedral angle is the plane bisecting the dihedral angle.

22.0 Between any two skew lines there is one, and only one, common perpendicular.

23.0 The sum of any two face angles of a trihedral angle is greater than the third face angle.

24.0 The sum of the face angles of any convex polyhedral angle is less than 360o.

25.0 If the three face angles of one trihedral angle are equal respectively to the three face angles of another, the corresponding dihedral angles of the trihedral angles are equal.

25.1 Two trihedral angles are either equal or symmetric if the three face angles of one are equal respectively to the three face angles of the other.

26.0 The lateral faces of a prism are parallelograms.

27.0 The lateral edges of a parallelogram are parallel and equal.

28.0 The lateral edges of a prism are perpendicular to the plane of a right section.

29.0 The lateral edges of a right prism are altitudes.

30.0 The lateral faces of a right prism are rectangles.

31.0 The bases of a prism are congruent.

31.1 Every section of a prism made by a plane parallel to the base is congruent to the base.

31.2 The sections of a prism made by parallel planes intersecting the lateral edges are congruent polygons.

32.0 Two prisms are congruent if the three faces which include a trihedral angle of one are congruent respectively to the three faces which include a trihedral angle of the other, and are similarly placed.

32.1 Two right prisms are congruent if they have congruent bases and equal altitudes.

33.0 Two truncated prisms are congruent if the three faces which include a trihedral angle of one are congruent respectively to the three faces which include a trihedral angle of the other, and are similarly placed.

34.0 An oblique prism is equal to a right prism whose base is a right section of the oblique prism and whose altitude is equal to a lateral edge of the oblique prism.

35.0 The lateral area of a prism is equal to the product of a lateral edge and the perimeter of a right section (S = ep).

35.1 The lateral area of a right prism is equal to the product of its altitude and the perimeter of its base.

36.0 All of the faces of a parallelepiped are parallelograms.

37.0 All of the faces of a rectangular solid are rectangles.

38.0 The opposite sides of a parallelepiped are parallel.

39.0 The opposite faces of a parallelepiped are congruent.

40.0 The faces of a cube are congruent squares.

41.0 The plane passed through two diagonally opposite edges of a parallelepiped divides it into two equal triangular prisms.

42.0 Cavalieri’s Theorem. Given two solids with equal altitudes included between two parallel planes. If every plane parallel to these two planes intersects both solids in cross-sections of equal area, then the two solids have equal volumes.

43.0 The volume of any parallelepiped is given by v = bh where b is the area of its base and h is its altitude.

44.0 The volume of a triangular prism is given by V = bh where b is the area of its base and h is its altitude.

44.1 The volume of any prism is given by V = bh where b is the area of its base and h is its altitude.

45.0 The bases of a cylinder are congruent.

45.1 Sections of a cylinder made by parallel planes cutting all elements are congruent.

45.2 A section of a circular cylinder made by a plane parallel to the plane of the base is a circle.

46.0 The lateral area of a circular cylinder is equal to the product of an element and the perimeter of a right section.

46.1 If S denotes the lateral area, T denotes the total area, h the altitude, and r the radius of the base of a right circular cylinder, then S = 2πrh, and T = 2πr(r + h).

47.0 The volume of a circular cylinder is given by V = bh where b is the area of the base and h is the altitude.

47.1 The volume of a circular cylinder is given by V = πr2h, where r is the radius and h is the altitude.

48.0 The lateral areas or the total areas of two similar cylinders of revolution have the same ratio as the squares of their altitudes or as the squares of the radii of their bases; and their volumes have the same ratio as the cubes of their altitudes or as the cubes of the radii of their bases.

49.0 The lateral edges of a regular pyramid are equal.

50.0 The lateral faces of a regular pyramid are congruent isosceles triangles.

51.0 The altitudes of the triangular faces of a regular pyramid are equal.

52.0 The lateral area of a regular pyramid is equal to half the product of its slant height and the perimeter of its base.

53.0 If a pyramid is cut by a plane parallel to the base, (a) the lateral edges and the altitude are divided proportionally, and (b) the section is similar to the base.

53.1 The area of a section of a pyramid parallel to the base is to the area of the base as the square of its distance from the vertex is to the square of the altitude of the pyramid.

53.2 If two pyramids have equal altitudes and equal bases, sections of the pyramids parallel to the bases and equidistant from the vertices are equal.

54.0 The lateral faces of a frustum of a pyramid are trapezoids.

54.1 The lateral faces of a frustum of a regular pyramid are congruent isosceles trapezoids.

54.2 The bases of a frustum of a pyramid are similar polygons.

55.0 the lateral area of a frustum of a regular pyramid is equal to half the product of its slant height and the sum of the perimeters of the bases.

56.0 Two pyramids having equal altitudes and equal bases are equal.

57.0 The volume of a triangular pyramid is given by V = ⅓bh where b is the area of its base and h is its altitude.

58.0 The volume of any pyramid is given by V = ⅓bh where b is the area of its base and h is its altitude.

58.1 The volume of a frustum of a pyramid is given by the formula

where b and b’ are the areas of the bases and h is the altitude.

59.0 The section of a circular cone made by a plane parallel to the base is a circle.

59.1 The area of a section of a circular cone made by a plane parallel to the base is to the area of the base as the square of its distance from the vertex of the cone is to the square of the altitude of the cone.

60.0 The lateral area of a right circular cone is equal to half the product of its slant height and the circumference of its base.

60.1 If S denotes the lateral area, T the total area, l the slant height, and r the radius of a right circular cone, then S = πrl, and T = πr(l + r).

60.2 If S denotes the lateral area of the frustum of a right circular cone, l the slant height, and r and rʹ the radii of the bases, then S = πl(r + rʹ).

60.3 The lateral area of the frustum of a right circular cone is equal to the circumference of a section midway between the bases, multiplied by the slant height.

61.0 The volume of a circular cone is given by V = ⅓bh, where b is the area of its base and h is its altitude.

61.1 The volume of a circular cone is given by the formula V = ⅓πr2h, where r is the radius of the base and h is the altitude.

61.2 The volume of the frustum of a circular cone is given by the formula V = ⅓πh(r2 + rʹ2 + rrʹ), where h is the altitude, and r and rʹ are the radii of the bases.

62.0 The lateral areas or the total areas of two similar cones of revolution have the same ratio as the squares of their altitudes, or as the squares of their slant heights, or as the squares of the radii of their bases; and their volumes have the same ratio as the cubes of their altitudes, or as the cubes of their slant heights, or as the cubes of the radii of their bases.

63.0 All radii of a sphere or of equal spheres are equal.

64.0 Spheres having equal radii are congruent.

65.0 All diameters of a sphere or of equal spheres are equal.

66.0 A point is within, on, or outside a sphere according as its distance from the center is less than, equal to, or greater than a radius.

67.0 A sphere may be generated by revolving a semicircle about the diameter as an axis.

68.0 If a plane intersects a sphere, the intersection is a circle.

68.1 The axis of a circle of a sphere passes through the center of the circle.

68.2 The center of a great circle of the sphere is the center of the sphere.

68.3 All great circles of a sphere are equal.

68.4 Any two great circles of a sphere bisect each other.

68.5 Through any two points on a sphere that are not the extremities of a diameter, one great circle of the sphere, and only one, can be drawn.

68.6 Through any three points on a sphere, one circle of the sphere, and only one, can be drawn.

69.0 The spherical distances of all points on a circle of a sphere from either pole of the circle are equal.

69.1 The polar distance of a great circle is a quadrant.

70.0 If a point on a sphere is at a distance of a quadrant from each of two other points on the sphere which are not the extremities of a diameter, then the point is a pole of the great circle passing through these two points.

See Fig. 1. Point P is at a distance of a quadrant from points A and B which are not the extremities of a diameter. Thus P is a pole of the great circle passing through points A and B.

71.0 If a plane is perpendicular to a radius at its extremity on the sphere, it is tangent to the sphere.

72.0 A plane tangent to a sphere is perpendicular to the radius drawn to the point of contact.

73.0 A sphere can be inscribed in a tetrahedron.

74.0 A cube can be circumscribed about a sphere.

75.0 A sphere can be passed through any four non-coplanar points.

76.0 A spherical angle is equal to the dihedral angle formed by the planes of its sides.

See Fig. 2. The spherical angle APB is equal to the dihedral angle A-PO-B. It is also equal to plane angle AOB.

77.0 A spherical angle is equal in degrees to the arc of the great circle which has the vertex of the angle as a pole and is included between the sides of the angle (extended if necessary).

See Fig. 2. The theorem states that spherical angle APB is equal, in degrees, to the intercepted arc AB.

78.0 The area of the surface generated by a straight line segment revolving about, but not crossing, an axis in its plane is given by S = 2πab where a is the length of the projection of the line segment on the axis and b is the length of the perpendicular from the axis to the line segment at its midpoint.

See Fig. 3. a = line segment CD and b = line segment EF where EF is the perpendicular bisector of AB; CD is the projection of AB on axis XY.

79.0 The area of a sphere with radius r is 4πr2.

79.1 The area of a sphere is equal to the product of its diameter and the circumference of a great circle.

79.2 The area of a sphere is equal to four times the area of one of its great circles.

79.3 The areas of two spheres are to each other as the squares of their radii or as the squares of their diameters.

80.0 The area of a zone is given by the formula A = 2πrh, where h is the altitude of the zone and r is the radius of the sphere.

81.0 If one of two great circles passes through the pole of the other, the circles are perpendicular to each other.

82.0 The volume of a sphere with radius r is 4/3 πr2 .

82.1 The volume of a sphere with diameter d is 1/6 πd3.

82.2 The volumes of two spheres have the same ratio as the cubes of their radii or as the cubes of their diameters.

83.0 If a cylinder is circumscribed about a sphere, the volume of the sphere is two thirds that of the cylinder, and the area of the sphere is equal to the lateral area of the cylinder, or two thirds the total area of the cylinder.

If a cone has its base and height equal respectively to the base and height of this cylinder, its volume is one third that of the cylinder, or one half that of the sphere.

84.0 The volume of a spherical sector or a spherical cone is given by V = 2/3 πr2h where r is the radius of the sphere and h is the altitude of the zone that is its base.

85.0 The volume of a spherical segment is given by V = 1/6 πh(h2 + 3r12 + 3 r22) , where h is the altitude and r1 and r2 are the radii of the bases.

86.0 The sides of a spherical polygon are equal in degrees to the corresponding face angles of the polyhedral angle.

See Fig. 4. Arc AB = ∠AOB, arc BC = ∠BOC, etc.

87.0 The angles of a spherical polygon are equal in degrees to the corresponding dihedral angles of the polyhedral angle.

See Fig. 4. ∠BAD = ∠B-OA-D, ∠ABC = ∠A-OB-C, etc.

88.0 Each side of a spherical triangle is less than the sum of the other two sides.

89.0 The sum of the sides of a spherical polygon is less than 360o.

90.0 The shortest path between two points on a sphere is the minor arc of the great circle through these points.

91.0 If one spherical triangle is the polar triangle of another, then the second is the polar triangle of the first.

92.0 In two polar triangles each angle of one is the supplement of the opposite side of the other.

This theorem states that in two polar triangles ABC and AʹBʹCʹ, ∠A and side BʹCʹ are supplementary, ∠B and side AʹCʹ are supplementary, and ∠C and side AʹBʹ are supplementary.

93.0 The sum of the angles of a spherical triangle is greater than 180o and less than 540o.

94.0 Two spherical triangles which are symmetric to a third triangle are congruent.

95.0 If two spherical triangles on a sphere or on equal spheres have two sides and the included angle of one equal respectively to two sides and the included angle of the other and arranged in the same order, the triangles are congruent.

95.1 Two symmetric isosceles spherical triangles are congruent.

96.0 If two spherical triangles on a sphere or on equal spheres have two angles and the included side of one equal respectively to two angles and the included side of the other and arranged in the same order, the triangles are congruent.

97.0 If two spherical triangles on a sphere or on equal spheres have two sides and the included angle of one equal respectively to two sides and the included angle of the other and arranged in the reverse order, the triangles are symmetric.

98.0 If two spherical triangles on a sphere or on equal spheres have two angles and the included side of one equal respectively to two angles and the included side of the other and arranged in the reverse order, the triangles are symmetric.

99.0 If two spherical triangles on a sphere or on equal spheres have the three sides of one equal respectively to the three sides of the other, the triangles are either congruent or symmetric.

99.1 If two sides of a spherical triangle are equal, the angles opposite those sides are equal.

100.0 If two spherical triangles on a sphere or on equal spheres have the three angles of one equal respectively to the three angles of the other, the triangles are either congruent or symmetric.

101.0 Two trihedral angles are either equal or symmetric if the three dihedral angles of one are equal respectively to the three dihedral angles of the other.

102.0 If two angles of a spherical triangle are equal, the sides opposite these angles are equal.

103.0 If two angles of a spherical triangle are unequal, the side opposite the greater angle is the greater.

104.0 If two sides of a spherical triangle are unequal, the angles opposite the greater side is the greater.

105.0 On a sphere or equal spheres the polar distances of equal circles are equal.

106.0 Two symmetric spherical triangles are equal.

107.0 The number of spherical degrees in the area of a lune is given by A = 2n, where n is the number of degrees in the angle of the lune.

108.0 The number of spherical degrees in the area of a spherical triangle is equal to the number of degrees in its spherical excess.

108.1 The ratio of the area of a spherical triangle to the area of the sphere is E/720, where E is the spherical excess of the triangle.

108.2 The number of spherical degrees in the area of a spherical polygon is equal to the number of degrees in its spherical excess.

109.0 The volume of a spherical pyramid is given by the formula V = πr3E/540, where r is the radius of the sphere and E is the spherical excess of the polygon.

110.0 The volume of a spherical wedge is given by the formula V = πr3A/270, where r is the radius of the sphere and A is the angle of the lune that is its base.

111.0 There can be no more than five regular polyhedrons.

112.0 Two tetrahedrons are similar if the three faces which include a trihedral angle of one are similar respectively to the three faces which include a trihedral angle of the other, and are similarly placed.

113.0 The volumes of two similar tetrahedrons have the same ratio as the cubes of any two corresponding sides.

114.0 Two similar polyhedrons can be separated into the same number of tetrahedrons, similar each to each and similarly placed.

114.1 Any two corresponding line segments of two similar polyhedrons have the same ratio as any two corresponding edges.

114.2 The areas of any two corresponding faces of two similar polygons have the same ratio as the squares of any two corresponding line segments.

114.3 The total areas of two similar polyhedrons have the same ratio as the squares of any two corresponding line segments.

114.4 The volumes of any two similar polyhedrons have the same ratio as the cubes of any two corresponding line segments.

115.0 If the number of edges of a polyhedron is denoted by E, the number of vertices by V, and the number of faces by F, then E + 2 = V + F.

116.0 The volume of a prismatoid is given by the formula V = 1/6 h(B1 + B2 + 4M), where h is the altitude, B1 is the area of the lower base, B2 is the area of the upper base, and M is the area of the midsection.

117.0 The volume of any truncated triangular prism is equal to one third the area of a right section multiplied by the sum of the lateral edges.