Question

determine if the following lines are parallel.

Explanation:

Step1: Recall parallel - line criteria

If two lines are cut by a transversal, and corresponding angles are equal, or alternate - interior angles are equal, or same - side interior angles are supplementary, the lines are parallel.

Step2: Analyze the first pair of lines

The two angles given are \(58^{\circ}\) and \(122^{\circ}\). Since \(58^{\circ}+ 122^{\circ}=180^{\circ}\), they are same - side interior angles. By the same - side interior angles supplementary criterion, the lines \(p\) and \(q\) are parallel.

Step3: Analyze the second pair of lines

The two angles marked as \(x^{\circ}\) are corresponding angles. Since corresponding angles are equal, the lines \(p\) and \(q\) are parallel.

Step4: Analyze the third pair of lines

The two \(60^{\circ}\) angles are corresponding angles. Since corresponding angles are equal, the lines \(p\) and \(q\) are parallel.

Answer:

For the first pair: Yes. Justification: Same - side interior angles are supplementary.
For the second pair: Yes. Justification: Corresponding angles are equal.
For the third pair: Yes. Justification: Corresponding angles are equal.