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Problem 138

If line \(\underline{\mathrm{AB}}\) is parallel to line \(\underline{\mathrm{CD}}\) and line \(\underline{\mathrm{EF}}\) is parallel to line \(\underline{\mathrm{GH}}\), prove that \(\mathrm{m}<1=\mathrm{m} \angle 2\).

Expert verified

To prove that \(\mathrm{m}<1=\mathrm{m} \angle 2\), we use the Alternate Interior Angles theorem which states that if two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. First, we identify that angles 1 and 3 are alternate interior angles between lines \(\underline{\mathrm{AB}}\) and \(\underline{\mathrm{CD}}\), so \(\mathrm{m}<1=\mathrm{m} \angle 3\). Then, we identify that angles 3 and 2 are alternate interior angles between lines \(\underline{\mathrm{EF}}\) and \(\underline{\mathrm{GH}}\), so \(\mathrm{m}<3=\mathrm{m} \angle 2\). Since \(\mathrm{m}<1=\mathrm{m} \angle 3\) and \(\mathrm{m}<3=\mathrm{m} \angle 2\), we can conclude that \(\mathrm{m}<1=\mathrm{m} \angle 2\), proving the statement.

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