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The aperture of a right circular cone is the maximum angle between two generatrix lines; if the generatrix makes an angle θ to the axis, the aperture is 2θ. In optics, the angle θ is called the half-angle of the cone, to distinguish it from the aperture.
Let two lines rotate about the points = (,) and = (,) so that when the line rotating about has angle with the x axis, the rotating about has angle . Let Q {\displaystyle Q} be the point of intersection, then the angle formed by the lines at Q {\displaystyle Q} is 2 θ {\displaystyle 2\theta } .
The angle between two planes (such as two adjacent faces of a polyhedron) is called a dihedral angle. [18] It may be defined as the acute angle between two lines normal to the planes. The angle between a plane and an intersecting straight line is complementary to the angle between the intersecting line and the normal to the plane.
A dihedral angle is the angle between two intersecting planes or half-planes. It is a plane angle formed on a third plane, perpendicular to the line of intersection between the two planes or the common edge between the two half-planes. In higher dimensions, a dihedral angle represents the angle between two hyperplanes.
Two lines are parallel if and only if the two angles of any pair of consecutive interior angles of any transversal are supplementary (sum to 180°). Proposition 1.28 of Euclid's Elements , a theorem of absolute geometry (hence valid in both hyperbolic and Euclidean Geometry ), proves that if the angles of a pair of consecutive interior angles ...
If two lines (a and b) are both perpendicular to a third line (c), all of the angles formed along the third line are right angles. Therefore, in Euclidean geometry, any two lines that are both perpendicular to a third line are parallel to each other, because of the parallel postulate. Conversely, if one line is perpendicular to a second line ...
Postulate III: Postulate of angle measure. The set of rays { ℓ, m, n , ...} through any point O can be put into 1:1 correspondence with the real numbers a (mod 2 π ) so that if A and B are points (not equal to O ) of ℓ and m , respectively, the difference a m − a ℓ (mod 2π) of the numbers associated with the lines ℓ and m is ∠ AOB .
The two great circles are shown as thin black lines, whereas the spherical lune (shown in green) is outlined in thick black lines. This geometry also defines lunes of greater angles: {2} π-θ, and {2} 2π-θ. In spherical geometry, a spherical lune (or biangle) is an area on a sphere bounded by two half great circles which meet at antipodal ...