Question

1. use the diagram below to answer the following questions.

a) name all segments parallel to (overline{xy})
b) name all segments parallel to (overline{zy})
c) name all segments parallel to (overline{yz})
d) name a plane parallel to plane (stu)
e) name a plane parallel to plane (uvz)
f) name all segments skew to (overline{sw})
g) name all segments skew to (overline{ut})

1. using the diagram below, describe the relationship as parallel, intersecting, or skew.

a) (overline{ab}) and (overline{bc})
b) (overline{ae}) and (overline{bf})
c) (overline{ef}) and (overline{ad})
d) plane (abc) and plane (abf)
e) plane (aed) and plane (bfc)

1. use the diagram below to answer the following questions.

f) name a transversal.
g) name all corresponding angles.
h) name all alternate interior angles.
i) name all alternate exterior angles.
j) name all consecutive interior angles.

1. use the diagram below to answer the following questions.

a) name the transversal connecting (angle1) and (angle5).
b) name the transversal connecting (angle7) and (angle14).
c) name the transversal connecting (angle8) and (angle11).
d) name the transversal connecting (angle6) and (angle15).
e) name the transversal connecting (angle3) and (angle9).

Explanation:

Step1: Recall parallel - segment concept

Parallel segments in 3 - D are segments that lie in the same direction and never intersect. For a cube - like figure in question 1:
a) Segments parallel to $\overline{XY}$ are $\overline{UV}$, $\overline{WZ}$, $\overline{ST}$.
b) Segments parallel to $\overline{ZY}$ are $\overline{VX}$, $\overline{UW}$, $\overline{TS}$.
c) Segments parallel to $\overline{VS}$ are $\overline{UW}$, $\overline{ZY}$, $\overline{XT}$.
d) A plane parallel to plane $STU$ could be plane $VWZ$.
e) A plane parallel to plane $UVZ$ could be plane $WXY$.
f) Skew segments to $\overline{SW}$ are segments that are not in the same plane and do not intersect. For example, $\overline{XY}$, $\overline{UV}$, $\overline{ZT}$, $\overline{XT}$, $\overline{ZY}$, $\overline{VX}$.
g) Skew segments to $\overline{UT}$ are $\overline{VX}$, $\overline{WZ}$, $\overline{XY}$, $\overline{SW}$, $\overline{SZ}$, $\overline{WY}$.

Step2: Recall line - line and plane - plane relationships

In question 2:
a) $\overline{AB}$ and $\overline{BC}$ are intersecting since they meet at point $B$.
b) $\overline{AE}$ and $\overline{BF}$ are parallel as they are in the same direction and do not intersect.
c) $\overline{EF}$ and $\overline{AD}$ are skew as they are not in the same plane and do not intersect.
d) Plane $ABC$ and plane $ABF$ intersect along line $\overline{AB}$.
e) Plane $AED$ and plane $BFC$ are parallel as they do not intersect.
f) In a set of parallel lines cut by a transversal, a transversal is a line that intersects two or more lines. For example, in question 3, a transversal could be the line that intersects the two parallel lines shown.
g) Corresponding angles are angles in the same relative position with respect to the transversal and the parallel lines. For example, if we have two parallel lines cut by a transversal, $\angle1$ and $\angle5$, $\angle2$ and $\angle6$, $\angle3$ and $\angle7$, $\angle4$ and $\angle8$ are corresponding angles.
h) Alternate interior angles are non - adjacent angles between the two parallel lines and on opposite sides of the transversal. For example, $\angle3$ and $\angle6$, $\angle4$ and $\angle5$.
i) Alternate exterior angles are non - adjacent angles outside the two parallel lines and on opposite sides of the transversal. For example, $\angle1$ and $\angle8$, $\angle2$ and $\angle7$.
j) Consecutive interior angles are angles between the two parallel lines and on the same side of the transversal. For example, $\angle3$ and $\angle5$, $\angle4$ and $\angle6$.

Step3: Identify transversals

In question 4:
a) The transversal connecting $\angle1$ and $\angle5$ is the line that intersects the lines containing $\angle1$ and $\angle5$.
b) The transversal connecting $\angle7$ and $\angle14$ is the relevant intersecting line.
c) The transversal connecting $\angle8$ and $\angle11$ is the intersecting line.
d) The transversal connecting $\angle6$ and $\angle15$ is the intersecting line.
e) The transversal connecting $\angle3$ and $\angle9$ is the intersecting line.

Answer:

1.
a) $\overline{UV}$, $\overline{WZ}$, $\overline{ST}$
b) $\overline{VX}$, $\overline{UW}$, $\overline{TS}$
c) $\overline{UW}$, $\overline{ZY}$, $\overline{XT}$
d) plane $VWZ$
e) plane $WXY$
f) $\overline{XY}$, $\overline{UV}$, $\overline{ZT}$, $\overline{XT}$, $\overline{ZY}$, $\overline{VX}$
g) $\overline{VX}$, $\overline{WZ}$, $\overline{XY}$, $\overline{SW}$, $\overline{SZ}$, $\overline{WY}$
2.
a) intersecting
b) parallel
c) skew
d) intersect along $\overline{AB}$
e) parallel
f) [appropriate line in the diagram]
g) [list of corresponding angles as described above]
h) [list of alternate interior angles as described above]
i) [list of alternate exterior angles as described above]
j) [list of consecutive interior angles as described above]
3.
a) [relevant transversal line]
b) [relevant transversal line]
c) [relevant transversal line]
d) [relevant transversal line]
e) [relevant transversal line]