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) another name for plane x is plane ☐.
(b) ☐ and f are distinct points that are collinear.
(c) point d and line ☐ are coplanar.
(d) suppose line \overleftrightarrow{dc} is drawn on the figure.
then \overleftrightarrow{dc} and \overleftrightarrow{☐} are distinct lines that intersect.

Explanation:

Part (a)

Step1: Recall plane naming rule

A plane can be named by three non - collinear points in it. Points \(D\), \(B\), \(E\) (or other combinations like \(B\), \(E\), \(F\), \(G\), \(D\)) lie in plane \(X\). Let's use \(B\), \(E\), \(D\) (or any three non - collinear points from \(D\), \(B\), \(E\), \(F\), \(G\)). So another name for plane \(X\) can be plane \(BED\) (or plane \(BEF\), plane \(BEG\), plane \(BDF\) etc. For example, using points \(B\), \(E\), \(D\))

Step2: Determine the answer

We can choose three non - collinear points in plane \(X\). Let's take \(B\), \(E\), \(D\). So another name for plane \(X\) is plane \(BED\) (other valid answers: plane \(BEF\), plane \(BEG\), plane \(BDF\) etc.)

Part (b)

Step1: Recall collinear points definition

Collinear points lie on the same line. From the figure, points \(E\), \(F\), \(G\) are on the same line. So a point that is collinear with \(F\) is \(E\) (or \(G\))

Step2: Determine the answer

We can choose \(E\) (or \(G\)). So the blank can be filled with \(E\) (or \(G\))

Part (c)

Step1: Recall coplanar definition

A point and a line are coplanar if the point lies in the plane that contains the line (or the line and the point lie in the same plane). Point \(D\) lies in plane \(X\). Line \(BE\) (or line \(BF\), line \(BG\), line \(BD\) etc.) lies in plane \(X\). Also, line \(AB\) (or line \(BC\), line \(AC\)) intersects plane \(X\) at \(B\), and since \(D\) is in plane \(X\), point \(D\) and line \(BE\) (or line \(AB\) etc.) are coplanar. Let's take line \(BE\) (or line \(AB\)). For example, line \(BE\) lies in plane \(X\) and \(D\) is in plane \(X\), so point \(D\) and line \(BE\) are coplanar. Another option is line \(AB\) (since \(B\) is in plane \(X\), \(D\) is in plane \(X\), so \(D\) and line \(AB\) (which passes through \(B\) in plane \(X\)) are coplanar)

Step2: Determine the answer

We can choose line \(BE\) (or line \(AB\), line \(BC\), line \(AC\), line \(BF\), line \(BG\), line \(BD\) etc.). So the blank can be filled with \(BE\) (or other valid lines as mentioned)

Part (d)

Answer:

Step1: Recall intersecting lines definition

Two distinct lines intersect if they have a common point. Line \(\overleftrightarrow{DC}\) passes through \(B\) (since \(B\) is on line \(AC\) and \(D\) is in plane \(X\)). Line \(\overleftrightarrow{BE}\) (or line \(\overleftrightarrow{BF}\), line \(\overleftrightarrow{BG}\), line \(\overleftrightarrow{BD}\)) passes through \(B\). So line \(\overleftrightarrow{DC}\) and line \(\overleftrightarrow{BE}\) (or line \(\overleftrightarrow{AB}\), line \(\overleftrightarrow{BC}\), line \(\overleftrightarrow{AC}\), line \(\overleftrightarrow{BF}\), line \(\overleftrightarrow{BG}\), line \(\overleftrightarrow{BD}\)) intersect at \(B\)

Step2: Determine the answer

We can choose line \(\overleftrightarrow{BE}\) (or line \(\overleftrightarrow{AB}\), line \(\overleftrightarrow{BC}\), line \(\overleftrightarrow{AC}\), line \(\overleftrightarrow{BF}\), line \(\overleftrightarrow{BG}\), line \(\overleftrightarrow{BD}\)). So the blank can be filled with \(\overleftrightarrow{BE}\) (or other valid lines as mentioned)

Final Answers

(a) \(\boldsymbol{BED}\) (or other valid three - point combination in plane \(X\))
(b) \(\boldsymbol{E}\) (or \(G\))
(c) \(\boldsymbol{BE}\) (or other valid line in plane \(X\) or passing through \(B\))
(d) \(\boldsymbol{\overleftrightarrow{BE}}\) (or other valid intersecting line)