Trigonometry angles of any triangle add up to 180 degrees. The two acute angles therefore add up to 90 degrees: they are complementary angles. The shape of a triangle is completely determined, except for similarity, by the angles. Once the angles are known, the ratios of the sides are determined, regardless of the overall size of the triangle. If the length of one of the sides is known, the other two are determined. These ratios are given by the following trigonometric functions of the known angle "A", where "a", " b" and "c" refer to the lengths of the sides in
Triangle inequality is an exterior angle of a right triangle and therefore larger than , meaning the other base angle of the isosceles triangle also is greater than and their sum exceeds in violation of the triangle postulate). This theorem establishing inequalities is sharpened by Pythagoras' theorem to the equality that the square of the length of the hypotenuse equals the sum of the squares of the other two sides. Consider a triangle whose sides are in an arithmetic progression and let the sides be , , . Then the triangle inequality requires that The first of these quadratic inequalities requires to