Pythagorean theorem straightedge and compass. Pythagoras's theorem enables construction of incommensurable lengths because the hypotenuse of a triangle is related to the sides by the square root operation. The figure on the right shows how to construct line segments whose lengths are in the ratio of the square root of any positive integer. Each triangle has a side (labeled "1") that is the chosen unit for measurement. In each right triangle, Pythagoras's theorem establishes the length of the hypotenuse in terms of this unit. If a hypotenuse is related to the unit by the square root of a positive integer that is
Sum of angles of a triangle answer was proven to be positive: in other spaces (geometries) this sum can be greater or lesser, but it then must depend on the triangle. Its difference from 180° is a case of "angular defect" and serves as an important distinction for geometric systems. In Euclidean geometry, the triangle postulate states that the sum of the angles of a triangle is two right angles. This postulate is equivalent to the parallel postulate. In the presence of the other axioms of Euclidean geometry, the following statements are equivalent: The sum of the angles of a hyperbolic triangle is less than 180°.