...construetion, also AB is common to the two triangles ABC and ADB, therefore the three sides of the one are equal to the three sides of the other, each to each. The angles -which are opposite to the equal sides in each triangle are equal. For, Produce the sides BC and BD...

...true, are not in all cases so ; one may be true, and the other false : thu», tin proposition, " If two triangles have the three sides of the one respectively equal to the three sides of the other, the three angles of the one will be respectively equal to the three augles of the other," may be proved...

...two spherical triangles * on the same sphere, or on equal spheres, have the three sides of the one equal to the three sides of the other, each to each, the angles also of the one shall be equal to the angles of the other, each to each, to which thevequal sides are...

...same manner, it may be proved, that AH is equal to DF ; hence the two triangles AGH, and DEF having the three sides of the one respectively equal to the three sides of the other are equal in every respecta ; and therefore, the angle at A equal to the angle at D, the angle AGH,...

...through two given points on the surface of the sphere. Hence, the three sides of the one triangle being equal to the three sides of the other, each to each, the triangles are equal. (Prop, vu.) Secondly, let the triangles be situated on contrary sides of the centre...

...the angle D, it must necessarily be greater. PROPOSITION XXV. THEOREM. Two triangles are equal which have the three sides of the one respectively equal to the three sides of the other. For the angle included between any two sides in the one triangle must be equal to the angle included...

...meet in the poles of that circle. PROP. V. If two spherical triangles have the three sides of the one equal to the three sides of the other, each to each, the angles which are opposite to the equal sides are likewise equal ; and conversely. PROP. VI. If two sides and...

...perpendicular to Ce; and join OQ. Then, because the triangles С P с, С Q с have the three sides of the one equal to the three sides of the other, each to each, the angle PC Ois equal (1. 7.) to the angle Q С О. And because (he triangles P С О, О С О have two...

...1.) is equal to the chord С В : and, because the triangles CDA, CDB have the three sides of the one equal to the three sides of the other, each to each, the angle CDA is equal to CDB : therefore, the line CD bisects AB at light angles. Therefore reversely,...

...equiangular triangle is also equilateral. PROP. VII. THEOR. If two triangles have the three sides of the one equal to the three sides of the other, each to each ; the angles opposite the equal sides are also equal. Let the two triangles ABC, DEF, have the three sides equal,...