Angle Properties Of Parallel Lines: A Second Method

          Previously, in "Angle Properties of Parallel Lines," I used congruent triangles, perpendicular to parallel lines, through their points of intersection with a third line to prove angle properties.  There is also a Cartesian method for determining these properties, involving the formula for the acute ∠ between two lines.  For interpretation I offer this illustration:

Fig. 7 Transversal Cutting Parallel Lines (Cartesian)

        Using the fact that the gradient of a line is equal to the tan value of the angle the line makes with the x-axis (m = tanθ), and the formula in trigonometry for the tangent of the difference of angles (tan(α - β)), we found a formula for the angle made between two lines (i.e. The angle that one line must be rotated through to coincide with the other).  That is...

                      θ = tan^-1((m₂ - m₁) / (1 + m₁m₂))

         For the proof, firstly we find the angle between an arbitrary line, y = a, and another (the transversal), y = mx + b (where m ≠ 0).  We then compare this to the angle between the transversal and a line parallel to the first, y = a + c (where c ≠ 0).

                Let α be the angle between the lines y = a and y = mx + b

                      α = tan^-1((m - 0) / (1 + 0.m))

                           = tan^-1(m)

                Let β be the angle between the lines y = a + c and 
                             y = mx + b

                      β = tan^-1((m - 0) / (1 + 0.m))

                           = tan^-1(m)

                      α = β

                ∴ where a transversal intersects a pair of parallel lines, corresponding ∠s are equal

         From here we can prove that alternate angles are equal (using vertically opposite ∠s) and that co-interior ∠s are supplementary (using supplementary ∠s on a straight line)