...arches of great circles, each of which is less than a semicircumference. PROP. III. (86.) 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...

...has been done in the analogous case of plane triangles (38). THEOREM. 482. y two triangles sitnated on the same sphere, or on equal spheres, are equilateral •with respect to each other, they will also be equiangular with respect to each other, and the equal angles wiU be opposite to equal...

...has been done in the analogous case of plane triangles (38). THEOREM. 482. If two triangles situattd on the same sphere, or on equal spheres, are equilateral with respect to each other, they will also be equiangular with respect to each other, and the equal angles will be opposite to equal...

...triangles A, B, the sides will be equal in the polar triangles P, Q, (476) ; but, since the triangles P, Q, are equilateral with respect to each other, they are also equiangular with respect to each other (482) ; and, the angles being equal in the triangles P, Q, it follows that the sides are equal in their...

...in the analogous case of plane triangles (38). THEOREM. 482. If two triangles situattd on the name sphere, or on equal spheres, are equilateral with respect to each other, they will also be equiangular with respect to each other, and the equal angles will be opposite to equal...

...wished to prove. 449. Theorem. If two spherical triangles on the same sphere, or on equal sphere?, are equilateral with respect to each other, they are also equiangular with respect to each other. Symmetrical Triangles. Demonstration. Let ABC, DBF (fig. 182) he the spherical triangles, of which...

...transposition, A>^-B1C — 360° + 180°, or *9>(5+ C) — 180°, as we wished to prove. 449. Theorem. If two spherical triangles on the same sphere, or...they are also equiangular with respect to each other. Proof. Let ABC, DEF (fig. 182) be the spherical triangles, of which the sides AB = DE, AC = DF, and...

...has been done in the analogous case of plane triangles (38). THEOREM. 482. If two triangles situated on the same sphere, or on equal spheres, are equilateral with respect to each other, they will also be equiangular with respect to each other, and the equal angles will be opposite to equal...

...transposition, A > -f- BC — 360° + 180°, or A>(B + C)— 180°, as we wished to prove. 449. Theorem. If two spherical triangles on the same sphere, or...they are also equiangular with respect to each other. Proof. Let ABC, DEF (fig. 182) be the spherical triangles, of which the sides AB = DE, AC = DF, and...

...sides of another, the two triangles are said to be equilateral with respect to each other. 457. If two triangles on the same sphere, or on equal spheres,...respect to each other, they are also equiangular with each other ; which follows from Arts. 440 and 369. 458. Place the centres of the equal spheres together,...