Langley’s Adventitious Angles Langley’s Adventitious Angles Langley’s Adventitious Angles is a mathematical problem posed by Edward Mann Langley in "The Mathematical Gazette" in 1922. In its original form the problem was as follows: formula_1 is an isosceles triangle. A solution was developed by James Mercer in 1923. This solution involves drawing one additional line, and then making repeated use of the fact that the internal angles of a triangle add up to 180° to prove that several triangles drawn within the large triangle are all isosceles. Many other solutions are possible. Cut the Knot list twelve different solutions and several alternative problems with
Special right triangle allows one to quickly calculate various lengths in geometric problems without resorting to more advanced methods. "Angle-based" special right triangles are specified by the relationships of the angles of which the triangle is composed. The angles of these triangles are such that the larger (right) angle, which is 90 degrees or radians, is equal to the sum of the other two angles. The side lengths are generally deduced from the basis of the unit circle or other geometric methods. This approach may be used to rapidly reproduce the values of trigonometric functions for the angles 30°, 45°, and 60°. Special