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Ingeometry, ananglesubtended (fromLatin for "stretched under") by aline segment at an arbitraryvertex is formed by the tworays between the vertex and eachendpoint of the segment. For example, aside of atrianglesubtends the opposite angle.
More generally, an angle subtended by anarc of acurve is the angle subtended by the correspondingchord of the arc.For example, acircular arcsubtends thecentral angle formed by the tworadii through the arc endpoints.
If an angle is subtended by a straight or curved segment, the segment is said tosubtend the angle. Sometimes the term "subtend" is applied in the opposite sense, and the angle is said tosubtend the segment. Alternately, the angle can be said tointercept orenclose the segment.
The above definition of a subtendedplane angle remains valid inthree-dimensional space (3D), as one vertex and two endpoints (assumed non-collinear) define anEuclidean plane in 3D.For example, an arc of agreat circle on asphere subtends a central plane angle, formed by the two radii between the center of the sphere and each of the two arc endpoints.
More generally, asurfacesubtends asolid angle if its boundary defines thecone of the angle.
Manytheorems in geometry relate to subtended angles. If two sides of a triangle arecongruent, then the angles they subtend are also congruent, and conversely if two angles are congruent then they are subtended by congruent sides (propositions I.5–6 inEuclid'sElements), forming anisosceles triangle. More generally, thelaw of sines states that thesine of each angle of a triangle is proportional to the side subtending it. Theinscribed angle theorem states that when the vertex of an angle inscribed in a circle lies on the same side of the chord subtending it as the center of the circle, then the central angle subtended by the same chord is twice the inscribed angle.
By extension, an angle subtended by a more complex geometric figure may be defined in terms of the figure'sconvex hull and itsdiameter; for example, the angle subtended by a tree as viewed in a camera (seeangular size).[1]A subtended plane angle can also be defined for any two arbitraryisolated points and a vertex, as in twolines of sight from a particular viewer; for example, the angle subtended by two stars as seen from Earth (seeangular separation).[2]
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