Question

exercises 1 - 4, consider the lines that contain the segments in the figure and the planes that contain the faces of the figure. all angles are right angles. which line(s) or plane(s) contain point a and appear to fit the description.

1. line(s) parallel to $overline{bf}$
2. line(s) perpendicular to $overline{bf}$

in exercises 5 - 8, use the diagram.

1. name a pair of parallel lines.
2. name a pair of perpendicular lines.
3. is $overline{ab}paralleloverline{bc}$? explain.
4. is $overline{bd}perpoverline{cd}$? explain.

in exercises 9 - 12, identify all pairs of angles of the given type.

1. alternate interior
2. alternate exterior
3. corresponding
4. consecutive interior

Explanation:

Step1: Recall parallel - line definition

Parallel lines are lines in the same plane that never intersect.

Step2: Analyze the first figure (rectangular - prism)

For the rectangular - prism, lines parallel to $\overline{BF}$ are $\overline{AE}$, $\overline{CG}$, $\overline{DH}$ since they have the same direction and will never intersect $\overline{BF}$.

Step3: Analyze skew - line definition

Skew lines are non - coplanar lines that do not intersect. In the rectangular - prism, lines skew to $\overline{BF}$ are $\overline{CD}$, $\overline{GH}$, $\overline{AD}$, $\overline{EH}$ etc. as they are not in the same plane as $\overline{BF}$ and do not intersect it.

Step4: Analyze perpendicular - line definition

Perpendicular lines intersect at a right angle. In the rectangular - prism, lines perpendicular to $\overline{BF}$ are $\overline{AB}$, $\overline{EF}$, $\overline{FG}$, $\overline{BC}$ etc. as they form right - angles with $\overline{BF}$.

Step5: Analyze parallel - plane definition

Planes parallel to plane $EFG$ are plane $ABCD$ and plane $DCHG$ as they will never intersect plane $EFG$.

Step6: Analyze the second figure (coordinate - like)

For the second figure, a pair of parallel lines could be $\overline{AC}$ and $\overline{BD}$ if they are constructed to be so. A pair of perpendicular lines could be $\overline{AC}$ and $\overline{AB}$ as they form a right - angle. $\overline{AB}$ is not parallel to $\overline{BC}$ because they intersect at point $B$. $\overline{BD}$ is not perpendicular to $\overline{CD}$ as they do not form a right - angle.

Step7: Analyze angle - pair definitions

For the third figure:

• Alternate interior angles: $\angle3$ and $\angle5$, $\angle4$ and $\angle6$.
• Alternate exterior angles: $\angle1$ and $\angle7$, $\angle2$ and $\angle8$.
• Corresponding angles: $\angle1$ and $\angle5$, $\angle2$ and $\angle6$, $\angle3$ and $\angle7$, $\angle4$ and $\angle8$.
• Consecutive interior angles: $\angle3$ and $\angle6$, $\angle4$ and $\angle5$.

Answer:

1. $\overline{AE}$, $\overline{CG}$, $\overline{DH}$
2. $\overline{CD}$, $\overline{GH}$, $\overline{AD}$, $\overline{EH}$ etc.
3. $\overline{AB}$, $\overline{EF}$, $\overline{FG}$, $\overline{BC}$ etc.
4. Plane $ABCD$, plane $DCHG$
5. $\overline{AC}$ and $\overline{BD}$ (example)
6. $\overline{AC}$ and $\overline{AB}$ (example)
7. No, they intersect at point $B$.
8. No, they do not form a right - angle.
9. $\angle3$ and $\angle5$, $\angle4$ and $\angle6$
10. $\angle1$ and $\angle7$, $\angle2$ and $\angle8$
11. $\angle1$ and $\angle5$, $\angle2$ and $\angle6$, $\angle3$ and $\angle7$, $\angle4$ and $\angle8$
12. $\angle3$ and $\angle6$, $\angle4$ and $\angle5$