...involving them) and also tor the third jf we should take, therefore, as in Geometry, two triangles which have two sides of one equal respectively to two sides of the other, and the included angles equal, then the respective equality of the remaining angles in each triangle...

...Conversely. If two triangles have two sides of the one respectively equal to two sides of the other, and the third side of the first greater than the third...side of the second, the included angle of the first will be greater than the included angle of the second. For, if the included angle of the first is not...

...Conversely. If two triangles have two sides of the one respectively equal to two sides of the other, and the third side of the first greater than the third...side of the second, the included angle of the first will be greater than the included angle of the second. For, if the included angle of the first is not...

...second. 2nd. Example. — A similar series of propositions occurs again in Theorems V. and XIII. " When two triangles have two sides of one equal respectively to two sides of the other," and — (a) " The included angle of the one equal to the included angle of the other, the base of one...

...triangle symmetrical to the first to be greater than the third side of the second triangle, and therefore the third side of the first greater than the third side of the second. 690. From 688 and 689, by 33, Rule of Inversion, if two spherical triangles have two sides of the one...

...triangle symmetrical to the first to be greater than the third side of the second triangle, and therefore the third side of the first greater than the third side of the second. 690. From 688 and 689, by 33, Rule of Inversion, if two spherical triangles have two sides of the one...

...BC. .-. BC > EF. QED Proposition 3O. Theorem. 120. Conversely, if two triangles have two sides of the one equal respectively to two sides of the other, but the third side of the first triangle greater than the third side of the second, then the included angle of the first triangle is...

...hypothenuse is equivalent to the sum of the polygons upon the other two sides. ADVANCE THEOREMS. 435. If two triangles have' two sides of one equal respectively to two sides of the other, and their included angles supplementary, the triangles are equivalent. 437. If through the middle point...

...until they meet, and find the areas of the two triangles of which the trapezoid is the difference. 171. If two triangles have two sides of one equal respectively to two sides of another and the included angles supplementary, the triangles are equal in area. PROPOSITION XI. THEOREM....

...ft. 3 in., what is the homologous altitude of the second ? 21. Two triangles are equivalent when they have two sides of one equal respectively to two sides of the other, and the included angles supplementary. 22. One diagonal of a rhomhus is five-thirds of the other, and...