Question

in the figure below, points d, b, e, f, and g lie in plane x. points a and c do not lie in plane x. for each part below, fill in the blanks to write a true statement. (a) point d and line are coplanar. (b) suppose line \\(\overrightarrow{dc}\\) is drawn on the figure. then \\(\overrightarrow{dc}\\) and are distinct lines that intersect. (c) another name for plane x is plane . (d) and f are distinct points that are collinear.

Explanation:

Step1: Recall coplanar concept

Coplanar points and lines lie in the same plane. Since points $D,B,E,F,G$ lie in plane $X$, a line formed by any of these points will be coplanar with point $D$. For example, line $\overleftrightarrow{EF}$ (any line formed by points in plane $X$ will work).

Step2: Identify intersecting lines

We know that line $\overleftrightarrow{DC}$ is drawn. A line that intersects it and is distinct can be a line in plane $X$ that passes through the intersection - point of $\overleftrightarrow{DC}$ and plane $X$ (if it intersects). Let's assume $\overleftrightarrow{AB}$ intersects $\overleftrightarrow{DC}$ at point $B$ (assuming they are in a position to intersect).

Step3: Rename the plane

A plane can be named by three non - collinear points in the plane. Since $D$, $E$, and $F$ are non - collinear points in plane $X$, we can name plane $X$ as plane $DEF$.

Step4: Recall collinear points

Collinear points lie on the same line. Points $E$ and $F$ lie on the same line in plane $X$, so $E$ and $F$ are distinct collinear points.

Answer:

(a) $\overleftrightarrow{EF}$ (any line formed by points $B,E,F,G$ will work)
(b) $\overleftrightarrow{AB}$
(c) $DEF$
(d) $E$