Question

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

Explanation:

Step1: Recall coplanar concept

Coplanar points and lines lie in the same plane. Since points $N,K,J,H$ lie in plane $Z$, we can choose a line formed by these points. For example, line $\overleftrightarrow{KJ}$ lies in plane $Z$, so point $N$ and line $\overleftrightarrow{KJ}$ are coplanar.

Step2: Name a plane by three non - collinear points

A plane can be named by three non - collinear points that lie in the plane. Points $N,K,H$ are non - collinear and lie in plane $Z$, so another name for plane $Z$ is plane $NKH$.

Step3: Find an intersecting line

We need to find a line that intersects $\overleftrightarrow{LM}$. Since $\overleftrightarrow{LM}$ is outside the plane $Z$ and we can consider a line that passes through the intersection of the plane containing $\overleftrightarrow{LM}$ and plane $Z$. Let's assume a line that intersects $\overleftrightarrow{LM}$ at a point outside plane $Z$. For simplicity, if we consider the three - dimensional space, we can think of a line that intersects $\overleftrightarrow{LM}$ at a point. Let's say a line that can be formed by extending a line segment from a point in plane $Z$ to a point on $\overleftrightarrow{LM}$. But a more straightforward answer could be a line in plane $Z$ that, if extended in 3 - D space, would intersect $\overleftrightarrow{LM}$. For example, if we consider the intersection of the plane containing $\overleftrightarrow{LM}$ and plane $Z$, we can find an intersecting line. Let's assume a line $\overleftrightarrow{JK}$ (assuming appropriate 3 - D orientation). $\overleftrightarrow{LM}$ and $\overleftrightarrow{JK}$ are distinct lines that intersect.

Step4: Recall collinear points

Collinear points lie on the same line. In plane $Z$, points $K,J,H$ are collinear (assuming they lie on a straight - line in the plane). So $K,J,H$ are distinct points that are collinear.

Answer:

(a) $\overleftrightarrow{KJ}$
(b) $NKH$
(c) $\overleftrightarrow{JK}$
(d) $J$, $H$