Question

determine which lines, if any, must be parallel. explain.
7.
8.

Explanation:

7.

Step1: Apply the perpendicular - parallel line theorem

If two lines are perpendicular to the same line, then they are parallel.
Line \(l\perp p\), \(l\perp q\), so \(p\parallel q\) (In a plane, if a line is perpendicular to one of two parallel lines, it is perpendicular to the other. Also, the converse: if two lines are perpendicular to the same line, they are parallel).
Line \(m\perp p\), \(m\perp q\), so \(p\parallel q\) (same theorem).
Line \(n\perp p\), \(n\perp q\), so \(p\parallel q\) (same theorem). Also, \(l\parallel m\) since \(l\perp p\) and \(m\perp p\), \(l\parallel n\) since \(l\perp q\) and \(n\perp q\), \(m\parallel n\) since \(m\perp p\) and \(n\perp p\)

Step1: Use the perpendicular - parallel line theorem

If two lines are perpendicular to the same line, then they are parallel.
Line \(a\perp c\), and there is no information to suggest \(b\) has a perpendicular relationship with \(c\) in the way that would make \(a\parallel b\) based on the perpendicular - parallel line theorem. However, if we assume the general case of the figure (where the right - angle for line \(a\) with \(c\) and the non - right - angle (visually, if we consider the standard geometric rules) for \(b\) and \(c\) is not enough. But if we consider the fact that \(a\perp c\) and if we assume the lines \(c\) and \(d\) are parallel (if we consider the two - line and transversal (the vertical lines) situation, but actually, since \(a\perp c\) and there is no indication that \(b\) has the same perpendicular relationship with \(c\) or \(d\) in a way to force parallelism. But if we consider the case of two lines (\(a\) and \(b\)) and two transversals (\(c\) and \(d\)). Since \(a\perp c\) and assume \(c\parallel d\) (if we consider the two horizontal lines as parallel in the geometric figure construction sense), and \(a\perp d\) (by the property of a line perpendicular to one of two parallel lines is perpendicular to the other). But for \(b\), if \(b\) is not shown to be perpendicular to \(c\) (or \(d\)) in the same way as \(a\). But if we assume the figure is constructed such that \(a\) and \(b\) are related to the two parallel lines \(c\) and \(d\). Since \(a\) is perpendicular to \(c\) and \(d\) (if \(c\parallel d\)), and if \(b\) is also related to \(c\) and \(d\) in the figure (visually, if we assume the non - right angle for \(b\) and \(c\) is a mistake in drawing and we go by the geometric rules: If two lines (\(a\) and \(b\)) are cut by two parallel transversals (\(c\) and \(d\)) and \(a\perp c\), \(a\perp d\), and if \(b\) is such that the corresponding angles (with respect to the transversals \(c\) and \(d\)) are equal (in the case of right angles for \(a\) and assuming \(b\) also forms right angles, but actually, from the given right - angle symbol for \(a\) and \(c\) only. But if we consider the general parallel line determination: If two lines are perpendicular to the same line (assuming \(c\) and \(d\) are parallel and \(a\) is a transversal perpendicular to \(c\) (and thus \(d\)), and if \(b\) is also a transversal that would be parallel to \(a\) if it were perpendicular to \(c\) (or \(d\)). But in the given figure, only \(a\perp c\) is marked. However, if we assume the two horizontal lines (\(c\) and \(d\)) are parallel (by the way the figure is drawn as two horizontal lines), and \(a\) is a vertical line (perpendicular to \(c\) and \(d\)). If \(b\) is also a vertical line (even though the right - angle for \(b\) and \(c\) is not marked, but in a standard geometric figure where two vertical lines (\(a\) and \(b\)) cut by two parallel horizontal lines (\(c\) and \(d\))), \(a\parallel b\) because they are both perpendicular to the parallel lines \(c\) and \(d\) (by the theorem: In a plane, if two lines are perpendicular to the same line (or in this case, to parallel lines, since if \(l_1\parallel l_2\) and \(m\perp l_1\), \(n\perp l_2\), and \(l_1\parallel l_2\), then \(m\parallel n\))

Answer:

\(l\parallel m\parallel n\) and \(p\parallel q\). Because in a plane, if two lines are perpendicular to the same line, then the two lines are parallel.

8.