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Problem 136

If \(\ell_{1} \| \ell_{2}\), prove that \(\angle 1\) is supplementary to $\angle 2$.

Expert verified

Let \(\ell_1\) and \(\ell_2\) be parallel lines and \(t\) be a transversal. Identify the corresponding angle to \(\angle 1\) (angle 3) and an alternate interior angle to \(\angle 2\) (angle 4). By corresponding angle property, \(\angle 1 = \angle 3\), and by alternate interior angle property, \(\angle 2 = \angle 4\). Since angle 3 and angle 4 form a linear pair, \(\angle 3 + \angle 4 = 180^\circ\). Substituting angles, we get \(\angle 1 + \angle 2 = 180^\circ\), proving angle 1 is supplementary to angle 2.

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