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
Degrees of freedom (statistics) degrees of freedom. In some complicated settings, such as unbalanced split-plot designs, the sums-of-squares no longer have scaled chi-squared distributions. Comparison of sum-of-squares with degrees-of-freedom is no longer meaningful, and software may report certain fractional 'degrees of freedom' in these cases. Such numbers have no genuine degrees-of-freedom interpretation, but are simply providing an "approximate" chi-squared distribution for the corresponding sum-of-squares. The details of such approximations are beyond the scope of this page. Several commonly encountered statistical distributions (Student's t, Chi-Squared, F) have parameters that are commonly referred to as "degrees of freedom". This terminology simply reflects that in many applications