Question

3-7 parallel lines and transversals
vocabulary term: definition / notation picture/diagram:
parallel lines
skew lines
parallel planes
number 1: identify each of the following using the cube shown. assume lines and planes that appear to be parallel or perpendicular are parallel or perpendicular, respectively.
a. all lines skew to $overleftrightarrow{bc}$
b. all lines parallel to $overleftrightarrow{eh}$
c. all planes parallel to plane $dch$
number 2: identify each of the following using the figure shown. assume lines and planes that appear to be parallel or perpendicular are parallel or perpendicular, respectively.
a. three segments parallel to $overline{ae}$ b. a segment skew to $overline{ab}$
c. a pair of parallel planes d. a segment parallel to $overline{ad}$
e. three segments parallel to $overline{hg}$ f. five segments skew to $overline{bc}$
g. how could you characterize the relationship between faces $abcd$ and $dcgh$? explain

Explanation:

Step1: Recall definitions

Parallel lines are lines in the same plane that never intersect. Skew lines are non - coplanar lines that do not intersect. Parallel planes are planes that do not intersect.

Step2: Analyze cube for Number 1

a. Lines skew to $\overline{BC}$ in cube $ABCDEFGH$

Lines that are not in the same plane as $\overline{BC}$ and do not intersect it are $\overline{AE}$, $\overline{DH}$, $\overline{EF}$, $\overline{HG}$.

b. Lines parallel to $\overline{EH}$

In the cube, lines parallel to $\overline{EH}$ are $\overline{FG}$, $\overline{AD}$, $\overline{BC}$.

c. Planes parallel to plane $DCH$

The plane parallel to plane $DCH$ is plane $ABF$.

Step3: Analyze figure for Number 2

a. Segments parallel to $\overline{AE}$

Segments parallel to $\overline{AE}$ are $\overline{BF}$, $\overline{CG}$, $\overline{DH}$.

b. Segment skew to $\overline{AB}$

A segment skew to $\overline{AB}$ could be $\overline{EH}$.

c. Pair of parallel planes

A pair of parallel planes could be plane $ABCD$ and plane $EFGH$.

d. Segment parallel to $\overline{AD}$

A segment parallel to $\overline{AD}$ is $\overline{BC}$.

e. Segments parallel to $\overline{HG}$

Segments parallel to $\overline{HG}$ are $\overline{EF}$, $\overline{AB}$, $\overline{CD}$.

f. Segments skew to $\overline{BC}$

Segments skew to $\overline{BC}$ are $\overline{AE}$, $\overline{DH}$, $\overline{EF}$, $\overline{HG}$, $\overline{EH}$.

g. Relationship between faces $ABCD$ and $DCGH$

The faces $ABCD$ and $DCGH$ share a common edge $\overline{DC}$. They are adjacent planes and are perpendicular to each other.

Answer:

Number 1:
a. $\overline{AE}$, $\overline{DH}$, $\overline{EF}$, $\overline{HG}$
b. $\overline{FG}$, $\overline{AD}$, $\overline{BC}$
c. Plane $ABF$
Number 2:
a. $\overline{BF}$, $\overline{CG}$, $\overline{DH}$
b. $\overline{EH}$
c. Plane $ABCD$ and plane $EFGH$
d. $\overline{BC}$
e. $\overline{EF}$, $\overline{AB}$, $\overline{CD}$
f. $\overline{AE}$, $\overline{DH}$, $\overline{EF}$, $\overline{HG}$, $\overline{EH}$
g. Adjacent and perpendicular, as they share edge $\overline{DC}$