Elements of geometry and trigonometry . y easy to obtain thedirect and absolute measure by this method ; since, oncomf)aririg tin* arc which serves as a nuMsure to any an-gle, with the fourth part of the cinMunfcrcnce, \\c. find theratio of the given angle to a right angle, which is the absolutemeasure. 54 GEOMETRY. Scholium 2. All that has been demonstrated in the last threepropositions, concerning the comparison of angles with arcs,holds true equally, if applied to the comparison of sectors witharcs ; for sectors are not only equal when their angles are so,but are in all respects proportiona
Elements of geometry and trigonometry . y easy to obtain thedirect and absolute measure by this method ; since, oncomf)aririg tin* arc which serves as a nuMsure to any an-gle, with the fourth part of the cinMunfcrcnce, \\c. find theratio of the given angle to a right angle, which is the absolutemeasure. 54 GEOMETRY. Scholium 2. All that has been demonstrated in the last threepropositions, concerning the comparison of angles with arcs,holds true equally, if applied to the comparison of sectors witharcs ; for sectors are not only equal when their angles are so,but are in all respects proportional to their angles : hence, twosectors ACB, ACD, taken in the same circle, or in equal circles,are to each other as the arcs AB, AD, the bases of those is hence evident .nat the arcs of the circle, which serve as ameasure of the different angles, are proportional to the differentsectors, in the sxmc circle, or in equal circles. PROPOSITION XVIII. THEOREM. An ribed angle is measured by half the arc included betiveen its Let BAD be an inscribed angle, and letus first suppose that the centre of the cir-cle lies within the angle BAD. Draw thediameter AE, and the radii CB, CD. The angle BCE, being exterior to thetriangle ABC, is equal to the sum of thetwo interior angles CAB, ABC (Book XXV. Cor. 6.) : but the triangle BACbeing isosceles, the angle CAB is equal toABC : hence the angle BCE is double of BAC. BCE liesat the centre, it is measured by the arc BE ; hence BAC will bemeasured by the half of BE. For a like reason, the angle CADwill be measured by the half of ED ; hence BAC + CAD, or BADwill be measured by half of BE + ED, or of BED. Suppose, in the second place, that thecentre C lies without the angle BAD. Thendrawing the diameter AE, the angle BAEwill be measured by the half of BE ; theangle DAE by the half of DE : hence theirdifference BAD will be measured by thehalf of BE minus the half of ED, or by thehalf of BD. Hence every inscribed angl
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