Question

in exercises 2 and 3, determine which lines, if any, must be parallel. explain your reasoning.
2.
3.

Explanation:

Exercise 2

Step1: Apply the perpendicular - parallel theorem

If two lines are perpendicular to the same line, then they are parallel. Line \(e\) is perpendicular to line \(g\) and line \(e\) is perpendicular to line \(h\).
Since \(e\perp g\) and \(e\perp h\), by the theorem "If two lines are perpendicular to the same line, then the two lines are parallel", we have \(g\parallel h\).

Exercise 3

Step1: Use the perpendicular - parallel theorem for \(j\) and \(k\)

Line \(j\) is perpendicular to line \(n\) and line \(k\) is perpendicular to line \(n\).
Since \(j\perp n\) and \(k\perp n\), by the theorem "If two lines are perpendicular to the same line, then the two lines are parallel", we have \(j\parallel k\).

Step2: Use the perpendicular - parallel theorem for \(l\) and \(n\)

Line \(l\) is perpendicular to \(j\) and line \(n\) is perpendicular to \(j\).
Since \(l\perp j\) and \(n\perp j\), by the theorem "If two lines are perpendicular to the same line, then the two lines are parallel", we have \(l\parallel n\).

Step3: Use the perpendicular - parallel theorem for \(m\) and \(n\)

Line \(m\) is perpendicular to \(k\) and line \(n\) is perpendicular to \(k\).
Since \(m\perp k\) and \(n\perp k\), by the theorem "If two lines are perpendicular to the same line, then the two lines are parallel", we have \(m\parallel n\).
Since \(l\parallel n\) and \(m\parallel n\), by the transitive property of parallel lines (\(a\parallel c\) and \(b\parallel c\) implies \(a\parallel b\)), we have \(l\parallel m\).

Answer:

For Exercise 2: \(g\parallel h\).
For Exercise 3: \(j\parallel k\), \(l\parallel m\), \(l\parallel n\), \(m\parallel n\).