Suppose that \(\ell, m, n\) are lines, such that \(n\) cuts \(\ell\) and \(m\) at unique points. We are given that \(\beta\) and \(\gamma\) form a linear pair, and our task is to prove that \(\ell \parallel m.\) Assume, for the sake of contradiction, that \(\ell \not\parallel m.\) Then \(\ell, m, n\) form a triangle.
But knowing the sum of interior angles is \(180^\circ,\) and \(\beta + \gamma = 180^\circ,\) that means \(\alpha = 0^\circ.\) This is absurd, because no triangle can have an angle with zero measure. Therefore, it must be that our original assumption was wrong, and thus \(\ell \parallel m\) after all. \(\blacksquare\)