  Lobachevsky's geometry abounds in surprising theorems (many of which had already been found by Saccheri). Here are a few: The three interior angles of a triangle add up to less than two right angles. The difference or “defect” is proportional to the triangle's area. Hence, in Lobachevskian geometry, similar triangles are congruent. Moreover, if a triangle is divided into smaller triangles, the defect of the whole equals the sum of the defects of the parts. Since the defect cannot be greater than two right angles, the area of triangles has a finite maximum. If a quadrilateral, by construction, has three right angles, the fourth angle is necessarily acute. Thus, in Lobachevskian geometry there are no rectangles.