...and join GH. fa/ V Then the triangles GBH, DEF are equal, since two sides and the included angle in the one are respectively equal to two sides and the included angle in the other ; hence, and by hypothesis, AB : BC : : GB : BH, that is, the sides BA, BC of the triangle...

...the equal angles in the second, are called homologous. 51. Theorem. Two triangles are equal, when two sides and the included angle of the one are respectively equal to two sides and the included angle of the other. Demonstration. In the two triangles ABC, D£JP(ng.31), let the angle A be equal to the...

...respect equal. They are proved to be such in the fourth proposition, u In , tivo sides and the contaHied angle of the one are respectively equal to two sides and the contained angle of the other : in the eighth, when the three sides of the one are equal to the three...

...consequently they are equal. (Ax. 13.) Hence, universally, Any two triangles are every way equal, when two sides and the included angle of the one, are respectively equal to two sides and the included angle of the other. PROPOSITION VI. THEOREM. If two triangles ABC, DEF, have two angles and the interjacent...

...has been proved (Prop. VIII.) that two triangles are equal, when two sides and the included angle in the one, are respectively equal to two sides and the included angle in the oilier ; we may now infer further, that two triangles are equivalent or equal in surface, when...

...respect equal. They are proved to be such in the fourth proposition, when two sides and the contained angle of the one are respectively equal to two sides and the contained angle of the other: in the eighth, when the three sides of the one are equal to the three...

...the equal angles in the second, are called homologous. 51. Theorem. Two triangles are equal, when two sides and the included angle of the one are respectively equal to two sides and the included angle of the other. Proof. In the two triangles ABC, DEF, (fig. 31), let the angle Jl be equal to the angle...

...and a increase or decrease together. Cor. 2. Two spherical triangles are equal : (537) 1°. When two sides and the included angle of the one are respectively equal to the two sides and the included angle of the other ; and therefore, FOR A SIDE. 265 Fig. 120a. 2°....

...to DF, and join GH. Then the triangles AGH, DEF are equal, since two sides and the included angle in the one, are respectively equal to two sides and the included angle in the other (Prop. VI., BI). But, by hypothesis, AB : DE : : AC : DF; therefore AB : AG : : AC : AH;...

...than CB ; then take AD equal to BC, and join DJB. Now, in the two triangles DAB and CBA, we have ^ A equal to CB by construction, the side AB common, and...CB and CA, and therefore the triangle is isosceles. 'IPROPOSITION VIII. .THEOREM. When two triangles have the three sides of the one respectively equal...