Question

given: ∠5 and ∠16 are supplementary, ∠2 ≅ ∠18
prove: q || r

Explanation:

Step1: Use the given information

Given that \(\angle5\) and \(\angle16\) are supplementary (\(\angle5+\angle16 = 180^{\circ}\)), and \(\angle2\cong\angle18\) (so \(\angle2=\angle18\)).

Step2: Use vertical - angle and corresponding - angle relationships

We know that \(\angle5\) and \(\angle10\) are vertical angles, so \(\angle5=\angle10\). Since \(\angle5+\angle16 = 180^{\circ}\), then \(\angle10+\angle16 = 180^{\circ}\). Also, \(\angle16\) and \(\angle18\) are supplementary (\(\angle16+\angle18 = 180^{\circ}\)) because they form a linear pair. Since \(\angle2=\angle18\) and \(\angle2\) and \(\angle10\) are corresponding angles.

Step3: Apply the converse of the corresponding - angles postulate

If two lines are cut by a transversal and the corresponding angles are equal, then the lines are parallel. Here, when we consider the transversal and lines \(q\) and \(r\), since \(\angle10=\angle18\) (because \(\angle5+\angle16 = 180^{\circ}\), \(\angle16+\angle18 = 180^{\circ}\) so \(\angle5=\angle18\) and \(\angle5 = \angle10\)), by the converse of the corresponding - angles postulate, \(q\parallel r\).

Answer:

By using the properties of supplementary angles, vertical angles, and the converse of the corresponding - angles postulate, we have proved that \(q\parallel r\).