A comparison theorem is a test of whether or not a mathematical object satisfies a set of predetermined properties. In calculus, a comparison can test if an integral is convergent or divergent. It is useful when one is not concerned with the actual value of an integral.
The lengths of the sides of a 30-60-90 triangle always exist in the proportional pattern 1:2:sqrt 3. The shorter side is half as long as the hypotenuse, and the length of the longer side is found by multiplying the length of the shorter side by the square root of three. Knowing this pattern makes it
Four different methods for determining triangle congruency by examining sets of sides and angles exist: SSS (side, side, side), SAS (side, angle, side), ASA (angle, side, angle) and AAS (angle, angle, side). The hypotenuse and leg method can also determine the congruency of right triangles.
The corollary to the triangle sum theorem is the law of cosines, better known as the Pythagorean theorem. This theorem is often used in calculus courses.
The work-energy theorem is a generalized description of motion that states that the work done by the sum of all forces acting on an object is equal to the change in that object's kinetic energy. This principle of work and its relationship to kinetic energy is a core mechanical physics concept.
The angle bisector theorem states that a line bisecting an angle in a triangle divides the side opposite the angle into two line segments that have lengths proportional to the lengths of the other sides. An angle bisector is a line that divides an angle into two equal angles; it is often depicted as
The base angle theorem says if two sides of a triangle are congruent, then the angles opposite those sides must also be congruent. The converse is also true. If two angles of a triangle are congruent, then the opposite sides must be also be congruent.
Examples of the squeeze theorem, g(x) ? f(x) ? h(x), show that if f(x) is always greater than g(x) and if f(x) is always less than h(x), then when g(x) is equal to h(x), f(x) must also be equal. Since f(x) always squeezes between g(x) and h(x), it must be equal when g(x) and h(x) are equal.
A linear pair of angles is always supplementary. This means that the sum of the angles of a linear pair is always 180 degrees. This is called the linear pair theorem.
The exterior angle theorem states that an angle exterior to a triangle equals the sum of the two angles not adjacent to it. The theorem’s basis derives from the fact that the sum of all the interior angles in a triangle equals 180 degrees.