...not further develop Lobatchevsky's idea. Among other things, he proves that "if in any rectilinear triangle the- sum of the three angles is equal to two right angles, so is this also the case for every other triangle," that is to say, each instance is a sample of the...

...the sum of the angles of the triangle ADC must be 2^ + 0, which is absurd (by I. above). IV. If in a triangle the sum of the three angles is equal to two right angles, a quadrilateral can always be constructed with four right angles and four equal sides exceeding in...

...equidistant from one another." One of Legendre's alternatives is, " There exists a triangle in which the sum of the three angles is equal to two right angles." One of the latest attempts to suggest a substitute is that of the Italian Ingrami (1904), " Two parallel...

...third side. 9. In an isosceles triangle, the angles opposite the equal sides are equal. 10. In any triangle, the sum of the three angles is equal to two right angles, or 180°. 11. If two angles of a triangle are given, the third may be found by subtracting their sum...

...jingle in a semicircle is a right angle, was in a position, first, to show that in any right-angled triangle the sum of the three angles is equal to two right angles, and then, by drawing the perpendicular from a vertex of any triangle to the opposite side and so dividing...

...angle in a semicircle is a right angle, was in a position, first, to show that in any right-angled triangle the sum of the three angles is equal to two right angles, and then, by drawing the perpendicular from a vertex of any triangle to the opposite side and so dividing...

...reaction to various agents, and so on. Similarly, in the case of triangles, we regard the fact that the sum of the three angles is equal to two right angles as connected with the bare trilateralness of the figure, but not connected with the relative lengths...

...spherical triangles in general. Hence spherical trigonometry is not dependent upon whether in a rectilineal triangle the sum of the three angles is equal to two right angles or not. 38. "We will now consider anew the right-angled rectilineal triangle ABC (Fig. 3I}, in which...

...parallels, it will intersect the other also (Proclus, fifth century AD); There exists a triangle in which the sum of the three angles is equal to two right angles (Legendre, 1752-1833); Given any three points not in a straight line, there exists a circle passing...

...property of one or several quantities at once a theorem. So if someone desires to demonstrate that in every triangle the sum of the three angles is equal to two right angles, they will call such a demonstration a theorem, because it does not demand or teach to construct a triangle...