Points, Lines, and Planes: Practice

The line-in-plane foundation applies because two distinct points of the line lie in the plane. It gives containment of the entire line, including beyond $A$ and $C$.

Opposite rays share an endpoint and extend in opposite directions along the same line. From $E$, point $D$ is in one direction and $F$ is in the other.

Three noncollinear points name the unique plane containing them. A line cannot contain all three, because they are noncollinear.

Their intersection is a line because the planes are distinct and meet.

Both $S$ and $T$ lie on that line, so the unique-line foundation identifies it as $\overleftrightarrow{ST}$.

Exactly one line contains two distinct points. Both names therefore refer to that unique line through $P$ and $Q$.

In the rectangular front face, $AB$ and $BF$ meet at a given square corner, so they are perpendicular.

Lines $AB$ and $EF$ are opposite edge lines of the front face, and $AB$ and $CD$ are opposite edge lines of the bottom face, so both pairs are parallel by the givens.

For $AB$ and $FG$, the sole candidate for a common plane is the front-face plane through noncollinear $A,B,F$.

It excludes $G$, which lies in the parallel back-face plane, so those lines are skew.

Distinct planes with no common point are parallel. The word skew classifies lines, not planes.

Their membership in $\mathcal P$ makes them coplanar. The stated absence of one line through all four means they are not all collinear.

A perspective drawing can show a projected crossing of lines that do not meet in space. The stated skew relationship means the actual lines are noncoplanar and disjoint.

One point lies on many lines. To use two-point line notation, name a second point different from $X$. A lowercase label such as $t$ can still name the given line.

The containment foundation requires two distinct points of the line in the plane.

One shared point allows a line that passes through the plane without lying in it.

Their common line can belong to many planes. The unique-plane foundation needs a third point off the line, not merely a third distinct point on it.