Geometry worksheet on identifying parallel, perpendicular, and skew lines and planes in a rectangular prism.

Problem Analysis

The image depicts a rectangular prism, and the task involves identifying relationships between lines and planes in the prism. We will solve each part step by step.

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Step-by-Step Solution

#### 9. Identify whether $\overrightarrow{AE}$ and $\overrightarrow{DH}$ are parallel, perpendicular, or skew.
- Analysis:
- $\overrightarrow{AE}$ is a vertical line segment from $A$ to $E$.
- $\overrightarrow{DH}$ is a vertical line segment from $D$ to $H$.
- Both $\overrightarrow{AE}$ and $\overrightarrow{DH}$ are parallel because they are both vertical and lie in parallel planes (the front and back faces of the prism).

- Answer: $\overrightarrow{AE}$ and $\overrightarrow{DH}$ are parallel.

#### 10. Identify whether $\overrightarrow{EA}$ and $\overrightarrow{GF}$ are parallel, perpendicular, or skew.
- Analysis:
- $\overrightarrow{EA}$ is a horizontal line segment from $E$ to $A$.
- $\overrightarrow{GF}$ is a horizontal line segment from $G$ to $F$.
- Both $\overrightarrow{EA}$ and $\overrightarrow{GF}$ are parallel because they are both horizontal and lie in parallel planes (the bottom and top faces of the prism).

- Answer: $\overrightarrow{EA}$ and $\overrightarrow{GF}$ are parallel.

#### 11. Identify whether $\overrightarrow{GC}$ and $\overrightarrow{HG}$ are parallel, perpendicular, or skew.
- Analysis:
- $\overrightarrow{GC}$ is a vertical line segment from $G$ to $C$.
- $\overrightarrow{HG}$ is a horizontal line segment from $H$ to $G$.
- These two line segments are perpendicular because one is vertical and the other is horizontal, and they intersect at point $G$.

- Answer: $\overrightarrow{GC}$ and $\overrightarrow{HG}$ are perpendicular.

#### 12. Name 4 lines perpendicular to $\overrightarrow{EF}$.
- Analysis:
- $\overrightarrow{EF}$ is a horizontal line segment on the bottom face of the prism.
- Lines perpendicular to $\overrightarrow{EF}$ must be vertical lines. These include:
- $\overrightarrow{AE}$
- $\overrightarrow{BF}$
- $\overrightarrow{CG}$
- $\overrightarrow{DH}$

- Answer: $\overrightarrow{AE}$, $\overrightarrow{BF}$, $\overrightarrow{CG}$, $\overrightarrow{DH}$.

#### 13. Name 3 lines parallel to $\overrightarrow{EF}$.
- Analysis:
- $\overrightarrow{EF}$ is a horizontal line segment on the bottom face of the prism.
- Lines parallel to $\overrightarrow{EF}$ must also be horizontal. These include:
- $\overrightarrow{AB}$
- $\overrightarrow{CD}$
- $\overrightarrow{GH}$

- Answer: $\overrightarrow{AB}$, $\overrightarrow{CD}$, $\overrightarrow{GH}$.

#### 14. Name 3 lines skew to $\overrightarrow{EF}$.
- Analysis:
- Skew lines are lines that are neither parallel nor intersecting.
- Lines skew to $\overrightarrow{EF}$ can be any vertical lines not in the same plane as $\overrightarrow{EF}$. These include:
- $\overrightarrow{AE}$
- $\overrightarrow{DH}$
- $\overrightarrow{CG}$

- Answer: $\overrightarrow{AE}$, $\overrightarrow{DH}$, $\overrightarrow{CG}$.

#### 15. Name 2 parallel planes.
- Analysis:
- Parallel planes are planes that do not intersect.
- Two parallel planes in the prism are:
- The top face ($ABCD$)
- The bottom face ($EFGH$)

- Answer: Plane $ABCD$ and Plane $EFGH$.

#### 16. Name 2 perpendicular planes.
- Analysis:
- Perpendicular planes are planes that intersect at a right angle.
- Two perpendicular planes in the prism are:
- The front face ($ABFE$)
- The side face ($AEHD$)

- Answer: Plane $ABFE$ and Plane $AEHD$.

#### 17. How many planes does the prism have?
- Analysis:
- A rectangular prism has 6 faces, and each face is a plane.

- Answer: 6 planes.

#### 18. Name the intersection of planes $ABE$ and $BFG$. Explain.
- Analysis:
- Plane $ABE$ contains points $A$, $B$, and $E$.
- Plane $BFG$ contains points $B$, $F$, and $G$.
- The intersection of these two planes is the line segment $\overrightarrow{BG}$, which lies on both planes.

- Answer: The intersection is $\overrightarrow{BG}$.

#### 19. Is line $\overrightarrow{BG}$ in plane $CGF$? Explain why or why not.
- Analysis:
- Plane $CGF$ contains points $C$, $G$, and $F$.
- Line $\overrightarrow{BG}$ does not lie in plane $CGF$ because point $B$ is not in plane $CGF$.

- Answer: No, $\overrightarrow{BG}$ is not in plane $CGF$ because point $B$ is not in plane $CGF$.

#### 20. How many planes contain line $\overrightarrow{AB}$? Name them.
- Analysis:
- Line $\overrightarrow{AB}$ lies on the following planes:
- Plane $ABCD$ (top face)
- Plane $ABFE$ (front face)
- Plane $ABGH$ (side face)

- Answer: 3 planes: Plane $ABCD$, Plane $ABFE$, Plane $ABGH$.

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Final Answers

1. $\overrightarrow{AE}$ and $\overrightarrow{DH}$ are parallel.
2. $\overrightarrow{EA}$ and $\overrightarrow{GF}$ are parallel.
3. $\overrightarrow{GC}$ and $\overrightarrow{HG}$ are perpendicular.
4. Lines perpendicular to $\overrightarrow{EF}$: $\overrightarrow{AE}$, $\overrightarrow{BF}$, $\overrightarrow{CG}$, $\overrightarrow{DH}$.
5. Lines parallel to $\overrightarrow{EF}$: $\overrightarrow{AB}$, $\overrightarrow{CD}$, $\overrightarrow{GH}$.
6. Lines skew to $\overrightarrow{EF}$: $\overrightarrow{AE}$, $\overrightarrow{DH}$, $\overrightarrow{CG}$.
7. Parallel planes: Plane $ABCD$ and Plane $EFGH$.
8. Perpendicular planes: Plane $ABFE$ and Plane $AEHD$.
9. Number of planes in the prism: 6.
10. Intersection of planes $ABE$ and $BFG$: $\overrightarrow{BG}$.
11. Line $\overrightarrow{BG}$ is not in plane $CGF$.
12. Planes containing $\overrightarrow{AB}$: Plane $ABCD$, Plane $ABFE$, Plane $ABGH$.

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