...PROPOSITION XXI. 704. Theorem.—On the same sphere, or on equal spheres, two spherical triangles having two sides and the included angle of the one equal to two...and the included angle of the other, each to each, are equal, or symmetrical and equivalent. DEMONSTRATION. Let ABC and A'B'C' be two spherical triangles,...

...required triangle. For, all triangles constructed with the given data must be equal, since they would have two sides and the included angle of the one equal to two sides and the included angle of the other. \ QEF PROBLEM XI. One side of a triangle and its adjacent angles being given, to construct...

...triangles do you see that look equal ? " (Ans. ABE, ACD.) "Then we shall try to prove that these two have two sides and the included angle of the one, equal to two sides of the included angle of the other." This done, you say, " Now what other triangles look equal?" (Ans....

...falls between CA and CB, and CH will meet AB in some point, as H. Draw HE. The triangles HCB and HCE have two sides and the included angle of the one, equal to the corresponding parts of the other, whence HE = HB (?). Now AH + HE > AE but AH + HE = AH + HB =...

...similar. Take AG equal to DE, and AH to DF; also, join GH. Then the triangles AGH, DEF, having two sides and the included angle of the one equal to two sides and the included angle of the other, are equal throughout (Theo. XII, Book I). Now, by hypothesis, Hence, it follows that...

...DB < AC + CB. NOTE. — Have pupils give demonstration, prolonging BD instead of AD. THEOREM XII. If two triangles have two sides and the included angle of the one, respectively, equal to two sides and the included, angle of the other, they are equal throughout. The...

...three given ares (any two of which being supposed greater than the third}. (Cf. Euc. I. xxii.) (7). If two triangles have two sides and the included angle...and the included angle of the other, each to each, the triangles are equal in every respect. For, since two sides of given length intersecting at a given...

...CAG equal to the angle D (Post. 7) ; make AG equal to DE, and draw GC. Then the triangles AGC and DEF have two sides and the included angle of the one equal...and the included angle of the other, each to each ; consequently, GC is equal to EF (PV). Now, the point G may be without the triangle ABC, it may be...

...is less. Proposition XXVI. A Theorem. 63. If two triangles have two sides and the included angle of one equal to two sides and the included angle of the other, each to each, the other homologous parts are also equal, and the triangles are equal. Proposition XXVII. A Theorem....

...is less. Proposition XXVI. A Theorem. 63. If two triangles have two sides and the included angle of one equal to two sides and the included angle of the other, each to each, the other homologous parts are also equal, and the triangles are equal. Proposition XXVII. A Theorem....