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) another name for plane z is plane p. (b) n, □, and □ are distinct points that are collinear. (c) h, □, □, and □ are distinct points that are coplanar. (d) suppose line $overleftrightarrow{jp}$ is drawn on the figure. then $overleftrightarrow{jp}$ and $overleftrightarrow{□}$ are distinct lines that intersect.

Question

Accepted Answer

### Part (a)
A plane can be named by three non - collinear points on it or a single point (if we consider the notation where a plane can be named by a point in it, but more accurately, a plane is named by three non - collinear points). Since points $M$, $P$, $H$, $J$, $K$ lie in plane $Z$, another name for plane $Z$ can be plane $MPH$ (or any combination of three non - collinear points from $M$, $P$, $H$, $J$, $K$). But the given answer starts with $P$, so a correct answer could be plane $PMH$ (or other valid combinations). However, if we follow the idea of naming a plane by a point in it (a more basic naming in some contexts), and since $P$ is in plane $Z$, but actually, the correct way is to use three non - collinear points. Let's assume we use three points, say $M$, $P$, $H$. So another name for plane $Z$ is plane $MPH$ (or plane $PHJ$, plane $MJK$ etc. as long as the points are in plane $Z$).

### Part (b)
Points $N$, $M$, and $L$ are collinear because they lie on the same straight line (the vertical line in the figure). So the blanks can be filled with $M$ and $L$ (in either order).

### Part (c)
Points $H$, $M$, $P$, $J$ (or any other combination of points from $M$, $P$, $H$, $J$, $K$) are coplanar because they lie in plane $Z$. For example, $H$, $M$, $P$, $J$ are distinct points that are coplanar.

### Part (d)
Line $\overleftrightarrow{JP}$ lies in plane $Z$, and the line containing $M$, $L$, $N$ (let's call it line $LN$) intersects plane $Z$ at point $M$. Also, line $\overleftrightarrow{JP}$ and line $\overleftrightarrow{LN}$ (the line through $L$, $M$, $N$) intersect at point $M$ (wait, no, $JP$ is in plane $Z$ and $LN$ passes through $M$ which is in plane $Z$). Wait, actually, line $\overleftrightarrow{JP}$ and line $\overleftrightarrow{LM}$ (or $\overleftrightarrow{LN}$): the line $\overleftrightarrow{LN}$ (with points $L$, $M$, $N$) and line $\overleftrightarrow{JP}$ intersect at $M$? No, $M$ is on $LN$ and in plane $Z$, and $JP$ is in plane $Z$. Wait, maybe the line is $\overleftrightarrow{LM}$ (or $\overleftrightarrow{LN}$). So if we draw line $\overleftrightarrow{JP}$, then $\overleftrightarrow{JP}$ and $\overleftrightarrow{LN}$ (or $\overleftrightarrow{LM}$) are distinct lines that intersect.

### Final Answers (Filling the Blanks)
(a) Another name for plane $Z$ is plane $\boldsymbol{MPH}$ (or other valid combination of three non - collinear points in plane $Z$, e.g., $PHJ$, $MJK$ etc.)
(b) $N$, $\boldsymbol{M}$, and $\boldsymbol{L}$ are distinct points that are collinear.
(c) $H$, $\boldsymbol{M}$, $\boldsymbol{P}$, and $\boldsymbol{J}$ are distinct points that are coplanar (or other combination of points in plane $Z$).
(d) Suppose line $\overleftrightarrow{JP}$ is drawn on the figure. Then $\overleftrightarrow{JP}$ and $\boldsymbol{\overleftrightarrow{LN}}$ (or $\overleftrightarrow{LM}$) are distinct lines that intersect.

(Note: The answers are based on the geometric concepts of planes, collinear points, coplanar points, and intersecting lines. For part (a), a plane is named by three non - collinear points on it. For part (b), collinear points lie on the same line. For part (c), coplanar points lie on the same plane. For part (d), two lines intersect if they share a common point.)

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Explanation:

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Final Answers (Filling the Blanks)

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Part (a)

Part (b)

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Final Answers (Filling the Blanks)