...the line DF will coincide with AC, and EF with BC. THEOREM 16. If two triangles have three sides of one equal to the three sides of the other, each to each, thefe triangles are equal in every rcfpeft. — 8 E. 1, or 17 D. 1. Ci» For if the triangle DEF be...

...which are equal to them, are equal to one another. (2l6.) COR. 2. Hence, if two spherical triangles have the three sides of the one equal to the three sides of the other, or two sides and the included angle in the one, equal to two sides and the included angle, in the other,...

...FG ; but, by hypothesis, BC : EF : : AC : DF ; consequently FG = DF ; hence the triangles EGF, 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 construction, the triangle EGF is equiangular with the triangle...

...equal sides BC and EF. THEOREM. 45. In an isosceles triangle the angles opposite to the equal sides are equal. Demonstration. Let the side AB = AC (fig....sides of the one, equal to the three sides of the qther, each to each, namely, AD common to both, AB — AC, by hypothesis, and BD = DC, by construction...

...however dissimilar ; thus a circle may be equivalent to a square, a triangle to a rectangle, &c. 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...

...base at right angles. OF 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...

...their parts ; they are not different, therefore, but equal; and we say, universally, When two triangles have the three' sides of the one equal to the three sides of the other respectively, the angles will also be equal, respectively, and the two triangles will be equal in all...

...one and the same great 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...

...the three angles of the one equal to the three angles of the other, each to each, they shall likewise have the three sides of the one equal to the three sides of the othrr, each to each, viz. those which are opposite to the equal angles.* Let the spherical triangles...

...and С с ; draw P О 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 scribe two circles, and kt them cut one another in P; and from P draw PM perpendicular to А В : then...