Question

in the figure below, points m, p, h, j, and k lie in plane z.
points l and n do not lie in plane z.
for each part below, fill in the blanks to write a true statement.
(a) point p and line (overleftrightarrow{square}) are coplanar.
(b) suppose line (pn) is drawn on the figure.
then (pn) and (overleftrightarrow{square}) are distinct lines that intersect.
(c) another name for plane z is plane (square).
(d) (square) and j are distinct points that are collinear

Explanation:

Step1: Analyze part (a)

Coplanar means lying on the same plane. Point \(P\) is in plane \(Z\). Lines in plane \(Z\) can be considered. Line \(KJ\) (or \(KH\), \(JH\) etc. within plane \(Z\)) and point \(P\) are coplanar.

Step2: Analyze part (b)

If line \(\overleftrightarrow{PN}\) is drawn. Another line in the figure that intersects \(\overleftrightarrow{PN}\) (assuming intersection in the plane - like if we consider the intersection of \(\overleftrightarrow{PN}\) with the plane \(Z\) and a line in \(Z\)). For example, if we assume the intersection of \(\overleftrightarrow{PN}\) with plane \(Z\) at some point (say \(M\) if we extend), but more accurately, if we consider the line that forms the side of the plane (but since \(M\) is on the line \(NL\) which is not in plane \(Z\) except the intersection point. A better approach: If we consider the line \(KJ\) (in plane \(Z\)) and assume \(\overleftrightarrow{PN}\) intersects \(KJ\) (by extending). But more simply, if we consider the line \(ML\) (since \(M\) is on the intersection of the line \(NL\) and the plane \(Z\) - a better pair is \(\overleftrightarrow{PN}\) and \(\overleftrightarrow{ML}\) (they intersect at \(M\))

Step3: Analyze part (c)

Another name for plane \(Z\) can be named by three non - collinear points in the plane. For example, plane \(PJH\) (since \(P\), \(J\), \(H\) are non - collinear points in plane \(Z\))

Step4: Analyze part (d)

Collinear points are points that lie on the same line. \(J\) and \(H\) are collinear (they lie on line \(JH\)), \(H\) and \(K\) are collinear (they lie on line \(HK\))

Answer:

(a) \(KJ\) (b) \(\overleftrightarrow{ML}\) (c) \(PJH\) (d) \(H\) and \(K\) (or \(J\) and \(H\))