...Theorem. If 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...

...thus two circles having the same radius arc equal ; and two triangles having the three sides of the one equal to the three sides of the other, each to each, are also equal. 162. Two figures are similar, which have the angles of the one equal to the angles...

...BC : EF : : AC : DF; consequently FG = DF; hence the triangles EOF, DEF, have the three sides of the one equal to the three sides of the other, each to each ; they are therefore equal (43). But, by constr .ction, the triangle EGF is equiangular with the triangle...

...thus two circles having the same radius are equal ; and two triangles having the three sides of the one equal to the three sides of the other, each to each, are also equal. 162. Two figures are similar, which have the angles of the one equal to the angles...

...GEOMETRY. Book I. s Sup. PROP. IV. THEOREM. If two triangles, ABC and DEF, have the three sides of the one equal to the three sides of the other, each to each, viif. AB to DE, AC to DF, and BC to EF, the triangles are equal in every respect. Let AB be that side...

...circle, 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...

...equilateral triangles. JoinBD; then because the triangles В FD, В С D have the three sides of the one equal to the three sides of the other, each to each, the angle В FD is equal to the angle В С D (I. 7.), that is, to the angle of a regular pentagon...

...last circle in F ; and join FA, F В. Then, because the triangles ECD, FAB have the three sides of the one equal to the three sides of the other, each to each, (7.) they are equal in every respect, and the angle at A is equal to the angle at C. Therefore, &c....

...therefore, BAC is greater than EDF. PROPOSITION X. THEOREM. If two triangles have the three sides of the one equal to the three sides of the other, each to each, the three angles will also b« equal, each to each, and the triangles themselves will be equal. Let...

...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...