Isosceles triangle is acute, right or obtuse depends only on the angle at its apex. In Euclidean geometry, the base angles cannot be obtuse (greater than 90°) or right (equal to 90°) because their measures would sum to at least 180°, the total of all angles in any Euclidean triangle. Since a triangle is obtuse or right if and only if one of its angles is obtuse or right, respectively, an isosceles triangle is obtuse, right or acute if and only if its apex angle is respectively obtuse, right or acute. In Edwin Abbott's book "Flatland", this classification of shapes was used
Ailles rectangle between the two, with sides of lengths 2, 2 and (by the Pythagorean theorem) 2. The remaining space at the top of the rectangle is a right triangle with small angles of 15° and 75°. and sides of − 1, + 1 and 2. From the construction of the rectangle, it follows that and An alternative construction (also by Ailles) places a 30°–60°–90° triangle in the middle, flanked by two 45°–45°–90° triangles in the lower corners. The lower-left triangle has sides of 1, 1, and , while the lower-right triangle has sides of , , and . The 30°–60°–90° triangle