Question

1. your friend claims that there is only one line that can be drawn perpendicular to pq. is your friend correct? explain your reasoning. 8. determine which lines must be parallel. explain your reasoning.

Explanation:

Question 7

In a plane, there are infinitely many lines perpendicular to a given line segment \(\overline{PQ}\). For example, if we consider different points on \(\overline{PQ}\) (or in the plane around it), we can construct multiple perpendicular lines. The claim that there is only one is incorrect.

Lines \(w\) and \(x\) are both perpendicular to line \(b\). By the theorem that if two lines are perpendicular to the same line, then they are parallel to each other. Similarly, lines \(w\) and \(z\) are both perpendicular to line \(b\) (for \(w\)) and line \(a\) (for \(z\), but since \(a\) and \(b\) are parallel - assume from the figure's structure where perpendicular relationships are shown in a grid - like pattern). Wait, looking at the figure: lines \(w\) and \(x\): since \(w\perp b\) and \(x\perp b\), by the "if two lines are perpendicular to the same line, then they are parallel" theorem. Also, lines \(w\) and \(z\): \(w\perp b\) and \(z\perp b\) (assuming the right - angle marks are as per the standard perpendicular - parallel relations in such grid - like figures).

Answer:

No, your friend is not correct. There are infinitely many lines perpendicular to \(\overline{PQ}\) in a plane.

Question 8