Elements of geometry and trigonometry . PROPOSITION XIX. THEOREM. 7V*e anrrle formed hy two chords^ which intersect each other y ismeasured by half the sum of the arcs included ietuven its sides. 50 GEOJMETRY. Let AB, CD, be two chords intersectingeach other at E : then will the angleAEC, or DEB, be measured by half ofAC + DB. Draw AF parallel to DC : then willthe arc DF be equal to A( (Prop. X.) ;and the angle FAB equal to the angleDEB (Book I. Prop. XX, Cor. 3.). Butthe angle FAB is measured by half thearc FDB (); therefore, DEBis measured by half of FDB ; that is, by half of DB +
Elements of geometry and trigonometry . PROPOSITION XIX. THEOREM. 7V*e anrrle formed hy two chords^ which intersect each other y ismeasured by half the sum of the arcs included ietuven its sides. 50 GEOJMETRY. Let AB, CD, be two chords intersectingeach other at E : then will the angleAEC, or DEB, be measured by half ofAC + DB. Draw AF parallel to DC : then willthe arc DF be equal to A( (Prop. X.) ;and the angle FAB equal to the angleDEB (Book I. Prop. XX, Cor. 3.). Butthe angle FAB is measured by half thearc FDB (); therefore, DEBis measured by half of FDB ; that is, by half of DB + DF, orhalf of DB +AC. In the same manner it might be proved thatthe angle AED is measured by half of AFD + PROPOSITION XX. THEOREM. The angle formed by two secants, is measured by half the diffe-rence of the arcs included between its sides. Let AB, AC, be tw^o secants : thenwill the angle BAC be measured byhalf the difference of the arcs BECand DF. Draw DE parallel to AC : then willthe arc EC be equal to DF, and theangle BDE equal to the angle BDE is measured by half the arcBE ; hence, BAC is also measured byhalf the arc BE ; that is, by half thedifference of BEC and EC, or half thedifference of BEC and DF.
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