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Illustration of 3 equal dihedral angles. "Two dihedral angles have the same ratio as their plane angles."

Equal Dihedral Angles

Illustration of 3 equal dihedral angles. "Two dihedral angles have the same ratio as their plane angles."

Illustration of a dihedral angle bisected by a plane. "Every point in a plane which bisects a dihedral angle is equidistant from the faces of the angle."

Dihedral Angle With Bisector

Illustration of a dihedral angle bisected by a plane. "Every point in a plane which bisects a dihedral…

Illustration of polyhedral angle. "The opening of three or more planes which meet at a common point is called a polyhedral angle."

Polyhedral Angle

Illustration of polyhedral angle. "The opening of three or more planes which meet at a common point…

Illustration of two polyhedral angles that are equal.

Equal Polyhedral Angles

Illustration of two polyhedral angles that are equal.

Illustration of two symmetrical polyhedral angles.

Symmetrical Polyhedral Angles

Illustration of two symmetrical polyhedral angles.

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

Trihedral Angle

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

Illustration of a convex polyhedral angle. "The sum of the face angles of any convex polyhedral angle is less than four right angles."

Convex Polyhedral Angle

Illustration of a convex polyhedral angle. "The sum of the face angles of any convex polyhedral angle…

Illustration used to show that two trihedral angles are equal or symmetrical when the three face angles of the one are respectively equal to the three face angles of the other.

Equal trihedral angles

Illustration used to show that two trihedral angles are equal or symmetrical when the three face angles…

Illustration showing two prisms are equal if the three faces which include a trihedral angle of the one are respectively equal to three faces which include a trihedral angle of the other, and are similarly placed.

Equal Prisms

Illustration showing two prisms are equal if the three faces which include a trihedral angle of the…

Illustration showing that the volume of 2 triangular pyramids, having the same trihedral angle of the one equal to a trihedral angle of the other, are to each other as the products of the three edges of these trihedral angles.

Triangular Pyramids for Volume

Illustration showing that the volume of 2 triangular pyramids, having the same trihedral angle of the…

Illustration of a sphere with a polyhedral angle whose vertex is at the center of the sphere and whose face angles are measured by the sides of the polygon, whose dihedral angles have the same numerical measure as the angles of the polygon.

Polyhedral Angle in a Sphere

Illustration of a sphere with a polyhedral angle whose vertex is at the center of the sphere and whose…

Illustration of a sphere with a polyhedral angle.

Polyhedral Angle in a Sphere

Illustration of a sphere with a polyhedral angle.

Illustration of a parabola. "If a line PT is drawn from any point P of the curve, bisecting the angle between PF and the perpendicular from P to the directrix, every point of the line PT, except P, is without the curve."

Point on Parabola

Illustration of a parabola. "If a line PT is drawn from any point P of the curve, bisecting the angle…

Illustration of half of an ellipse. "If through a point P of an ellipse a line is drawn bisecting the angle between one of the focal radii and the other produced, every point in this line except P is without the curve."

Line Bisecting Angle Between Focal Radii on Ellipse

Illustration of half of an ellipse. "If through a point P of an ellipse a line is drawn bisecting the…

Illustration of a hyperbola with a line bisecting the focal radii. "If through a point P of an hyperbola a line is drawn bisecting the angle between the focal radii, every point in this line except P is on the convex side of the curve.

Line Bisecting Angle Between Focal Radii in Hyperbola

Illustration of a hyperbola with a line bisecting the focal radii. "If through a point P of an hyperbola…

Illustration of perpendicular lines. Intersecting lines that meet at a right angle.

Perpendicular Lines

Illustration of perpendicular lines. Intersecting lines that meet at a right angle.

Illustration of perpendicular lines with the horizontal and vertical lines labeled. "A horizontal line is a line parallel to the horizon, or water level. A vertical line is a line perpendicular to a horizontal line; consequently, it has the direction of a plumb-line."

Perpendicular Lines With Horizontal and Vertical Labeled

Illustration of perpendicular lines with the horizontal and vertical lines labeled. "A horizontal line…

Illustration that can be used to show the definition of an angle - an opening between two lines that intersect or meet.

Angle

Illustration that can be used to show the definition of an angle - an opening between two lines that…

Illustration that can be used to show supplementary angles. The angles form a straight line.

Supplementary Angles

Illustration that can be used to show supplementary angles. The angles form a straight line.

Illustration of angles with parallel corresponding sides.

Angles With Parallel Sides

Illustration of angles with parallel corresponding sides.

Illustration of angles with parallel sides going in the opposite direction.

Angles With Parallel Sides

Illustration of angles with parallel sides going in the opposite direction.

Illustration of angles with parallel sides. "GH is parallel to and lies in the same direction as DE, and HI is parallel to but lies in the opposite direction to EF; hence, angle GHI is the supplement of DEF."

Angles With Parallel Sides

Illustration of angles with parallel sides. "GH is parallel to and lies in the same direction as DE,…

Illustration of a circle with an inscribed angle.

Inscribed Angle in a Circle

Illustration of a circle with an inscribed angle.

"If AOB is an angle of 1 ° on the larger circle, it is also 1 ° on the smaller concentric circle, and the length of the arc AB is to the length of the arc CD as to radius OB is to the radius OD; or, arc AB:arc CD = OB:OD."

Concentric Circles With Angle of 1 °

"If AOB is an angle of 1 ° on the larger circle, it is also 1 ° on the smaller concentric circle,…

"In trigonometry, the arcs of circles are used to measure angles. All angles are supposed to have their vertexes at the center O of the circle, one side of the angle lying to the right of O, and coinciding with the horizontal diameter, as OB."

Arcs and Angles of a Trigonometric Circle

"In trigonometry, the arcs of circles are used to measure angles. All angles are supposed to have their…

Right triangle OCA, inside of Circle O is used to show that side AC is "opposite" O and side OC is "adjacent" to O. OA is the hypotenuse. Sine is defined as the ratio of the opposite side to the hypotenuse (AC/OA). Cosine is defined as the ratio of the adjacent side to the hypotenuse (OC/OA), and Tangent is defined as the ratio of the opposite side to the adjacent side (DB/OB).

Trigonometry Triangle to Show Sine, Cosine, and Tangent

Right triangle OCA, inside of Circle O is used to show that side AC is "opposite" O and side OC is "adjacent"…

The coversedsine is equal to 1 minus the sine. In this figure, AE is parallel to OB; hence, EO = AC = sine of angle AOC, when radius OA - 1. Therefore, Coversedsine = 1 - sine = 1 - AC/OA = 1 - AC = 1 - EO = ER, when OA = 1.

Versedsine and Coversedsine in Circle

The coversedsine is equal to 1 minus the sine. In this figure, AE is parallel to OB; hence, EO = AC…

Right triangle with angle of 29 ° 31' and side of 24.

Triangle With Angle of 29 ° 31' and Side of 24

Right triangle with angle of 29 ° 31' and side of 24.

Right triangle with angle of 25 ° 33' 7" and side of 37' 7"

Triangle With Angle of 25 ° 33' 7" and Side of 37' 7"

Right triangle with angle of 25 ° 33' 7" and side of 37' 7"

Oblique triangle with perpendicular drawn to form two right triangles, one with angle measuring 36 ° 3' 29" and sides 19" and 23 ".

Oblique Triangle With Perpendicular Drawn

Oblique triangle with perpendicular drawn to form two right triangles, one with angle measuring 36 °…

Oblique triangle with perpendicular drawn and angle of measure 35 ° 38" given, and sides of 88' 6" and 57'. This would be used with Law of Sines Ambiguous Case.

Oblique Triangle Used For Ambiguous Case

Oblique triangle with perpendicular drawn and angle of measure 35 ° 38" given, and sides of 88'…

Projections of a line perpendicular to a plane.

Projections of a Line

Projections of a line perpendicular to a plane.

Illustration of a rectangular bipyramid viewed at an angle. A bipyramid, or dipyramid, is formed by joining two congruent pyramids at their bases.

Rectangular Bipyramid

Illustration of a rectangular bipyramid viewed at an angle. A bipyramid, or dipyramid, is formed by…

Illustration of a right circular cone resting on an element such that the vertex is on the bottom and a vertical and horizontal element meet at a right angle.

Right Circular Cone on Side

Illustration of a right circular cone resting on an element such that the vertex is on the bottom and…

Illustration of a hollow cylinder viewed at an angle.

Hollow Cylinder

Illustration of a hollow cylinder viewed at an angle.

Illustration of a hollow cylinder viewed at an angle from above.

Hollow Cylinder

Illustration of a hollow cylinder viewed at an angle from above.

Illustration of an equilateral triangle that shows both the centroid (where the medians of the sides meet) and the incenter (where the angle bisectors meet).

Centers of Equilateral Triangle

Illustration of an equilateral triangle that shows both the centroid (where the medians of the sides…

Illustration showing the golden angle. The golden angle is the smaller of two angles created by dividing the circumference of a circle according to the golden section. The ratio of the length of the larger arc to the smaller arc is equal to the ratio of the entire circumference to the larger arc. The golden angle is approximately 137.51°.

Golden Angle

Illustration showing the golden angle. The golden angle is the smaller of two angles created by dividing…

Illustration of a polar graph/grid that is marked and labeled in 30° increments and units marked to 10.

Polar Grid In Degrees With Radius 10

Illustration of a polar graph/grid that is marked and labeled in 30° increments and units marked…

Illustration of a polar graph/grid that is marked, but not labeled, in 30° increments and units marked to 10.

Polar Grid In Degrees With Radius 10

Illustration of a polar graph/grid that is marked, but not labeled, in 30° increments and units…

Illustration of a polar graph/grid that is a unit circle marked and labeled in 30° increments.

Polar Grid In Degrees With Radius 1

Illustration of a polar graph/grid that is a unit circle marked and labeled in 30° increments.

Illustration of a polar graph/grid that is a unit circle marked, but not labeled, in 30° increments.

Polar Grid In Degrees With Radius 1

Illustration of a polar graph/grid that is a unit circle marked, but not labeled, in 30° increments.

Illustration of a polar graph/grid that is marked and labeled in 30° increments and units marked to 2.

Polar Grid In Degrees With Radius 2

Illustration of a polar graph/grid that is marked and labeled in 30° increments and units marked…

Illustration of a polar graph/grid that is marked, but not labeled, in 30° increments and units marked to 2.

Polar Grid In Degrees With Radius 2

Illustration of a polar graph/grid that is marked, but not labeled, in 30° increments and units…

Illustration of a polar graph/grid that is marked and labeled in 30° increments and units marked to 3.

Polar Grid In Degrees With Radius 3

Illustration of a polar graph/grid that is marked and labeled in 30° increments and units marked…

Illustration of a polar graph/grid that is marked, but not labeled, in 30° increments and units marked to 3.

Polar Grid In Degrees With Radius 3

Illustration of a polar graph/grid that is marked, but not labeled, in 30° increments and units…

Illustration of a polar graph/grid that is marked and labeled in 30° increments and units marked to 4.

Polar Grid In Degrees With Radius 4

Illustration of a polar graph/grid that is marked and labeled in 30° increments and units marked…

Illustration of a polar graph/grid that is marked, but not labeled, in 30° increments and units marked to 4.

Polar Grid In Degrees With Radius 4

Illustration of a polar graph/grid that is marked, but not labeled, in 30° increments and units…

Illustration of a polar graph/grid that is marked and labeled in 30° increments and units marked to 5.

Polar Grid In Degrees With Radius 5

Illustration of a polar graph/grid that is marked and labeled in 30° increments and units marked…

Illustration of a polar graph/grid that is marked, but not labeled, in 30° increments and units marked to 5.

Polar Grid In Degrees With Radius 5

Illustration of a polar graph/grid that is marked, but not labeled, in 30° increments and units…

Illustration of a polar graph/grid that is marked and labeled in 30° increments and units marked to 6.

Polar Grid In Degrees With Radius 6

Illustration of a polar graph/grid that is marked and labeled in 30° increments and units marked…

Illustration of a polar graph/grid that is marked, but not labeled, in 30° increments and units marked to 6.

Polar Grid In Degrees With Radius 6

Illustration of a polar graph/grid that is marked, but not labeled, in 30° increments and units…

Illustration of a polar graph/grid that is marked and labeled in 30° increments and units marked to 7.

Polar Grid In Degrees With Radius 7

Illustration of a polar graph/grid that is marked and labeled in 30° increments and units marked…

Illustration of a polar graph/grid that is marked, but not labeled, in 30° increments and units marked to 7.

Polar Grid In Degrees With Radius 7

Illustration of a polar graph/grid that is marked, but not labeled, in 30° increments and units…

Illustration of a polar graph/grid that is marked and labeled in 30° increments and units marked to 8.

Polar Grid In Degrees With Radius 8

Illustration of a polar graph/grid that is marked and labeled in 30° increments and units marked…

Illustration of a polar graph/grid that is marked, but not labeled, in 30° increments and units marked to 8.

Polar Grid In Degrees With Radius 8

Illustration of a polar graph/grid that is marked, but not labeled, in 30° increments and units…

Illustration of a polar graph/grid that is marked and labeled in 30° increments and units marked to 9.

Polar Grid In Degrees With Radius 9

Illustration of a polar graph/grid that is marked and labeled in 30° increments and units marked…

Illustration of a polar graph/grid that is marked, but not labeled, in 30° increments and units marked to 9.

Polar Grid In Degrees With Radius 9

Illustration of a polar graph/grid that is marked, but not labeled, in 30° increments and units…

Illustration of a polar graph/grid that is marked and labeled in 15° increments and units marked to 10.

Polar Grid In Degrees With Radius 10

Illustration of a polar graph/grid that is marked and labeled in 15° increments and units marked…

Illustration of a polar graph/grid that is marked, but not labeled, in 15° increments and units marked to 10.

Polar Grid In Degrees With Radius 10

Illustration of a polar graph/grid that is marked, but not labeled, in 15° increments and units…