Question

at the right, assume that all lines and planes that appear to be parallel are parallel.

1. name all lines parallel to cg.
2. what plane is parallel to plane dcgh?
3. name all lines skew to eh.
4. name all lines that are parallel to plane efgh.

5 - 10: use the figure at the right, assume that all lines and planes that appear to be parallel are parallel. determine if each statement is true or false.

1. statement about skew lines.
2. ag is parallel to plane hkl.
3. gl and cd are parallel lines.
4. plane djk is parallel to plane abh.
5. fl and ic are skew lines.
6. de is skew to plane agf.
7. use the figure at the right to complete each statement. ∠3 and ___ are same - side interior angles.

Explanation:

Step1: Recall parallel - line and plane definitions

Parallel lines in 3 - D are lines that lie in the same plane and do not intersect. Skew lines are non - coplanar lines that do not intersect. A line is parallel to a plane if it does not intersect the plane.

Step2: Analyze for parallel lines to $\overleftrightarrow{CG}$

In a rectangular prism, lines parallel to $\overleftrightarrow{CG}$ are $\overleftrightarrow{BF}$, $\overleftrightarrow{AE}$, $\overleftrightarrow{DH}$.

Step3: Analyze for plane parallel to plane $DCGH$

The plane parallel to plane $DCGH$ is plane $ABFE$.

Step4: Analyze for skew lines to $\overleftrightarrow{EH}$

Skew lines to $\overleftrightarrow{EH}$ are $\overleftrightarrow{AB}$, $\overleftrightarrow{BC}$, $\overleftrightarrow{CD}$, $\overleftrightarrow{DA}$, $\overleftrightarrow{BF}$, $\overleftrightarrow{AG}$.

Step5: Analyze for lines parallel to plane $EFGH$

Lines parallel to plane $EFGH$ are $\overleftrightarrow{AB}$, $\overleftrightarrow{BC}$, $\overleftrightarrow{CD}$, $\overleftrightarrow{DA}$.

Step6: Analyze statements 5 - 10

• For statement 5: Need to know which two lines are being referred to.
• For statement 6: $\overleftrightarrow{AG}$ is parallel to plane $HKL$ if the planes are oriented such that they do not intersect. Without seeing the full context of plane $HKL$ relative to the prism, assume based on parallel - plane and line - plane concepts. If the prism is oriented correctly, this can be true.
• For statement 7: $\overleftrightarrow{GL}$ and $\overleftrightarrow{CD}$ are not parallel as they are non - coplanar (skew lines). False.
• For statement 8: Need to analyze the orientation of planes $DJK$ and $ABH$. If they do not intersect and are in parallel positions, then plane $DJK$ is parallel to plane $ABH$.
• For statement 9: $\overleftrightarrow{FL}$ and $\overleftrightarrow{IC}$ are skew lines if they are non - coplanar and do not intersect.
• For statement 10: $\overleftrightarrow{DE}$ is skew to plane $AGF$ if $\overleftrightarrow{DE}$ does not lie in the plane $AGF$ and does not intersect it.

Step7: Analyze statement 11

Same - side interior angles are on the same side of a transversal and between two lines. Need to see the figure with $\angle3$ to determine the other angle.

1. Lines parallel to $\overleftrightarrow{CG}$: $\overleftrightarrow{BF}$, $\overleftrightarrow{AE}$, $\overleftrightarrow{DH}$
2. Plane parallel to plane $DCGH$: plane $ABFE$
3. Skew lines to $\overleftrightarrow{EH}$: $\overleftrightarrow{AB}$, $\overleftrightarrow{BC}$, $\overleftrightarrow{CD}$, $\overleftrightarrow{DA}$, $\overleftrightarrow{BF}$, $\overleftrightarrow{AG}$
4. Lines parallel to plane $EFGH$: $\overleftrightarrow{AB}$, $\overleftrightarrow{BC}$, $\overleftrightarrow{CD}$, $\overleftrightarrow{DA}$

5 - 10: Need more information about the specific elements in the second figure for definite true/false answers.
11: Need to see the figure with $\angle3$ for a definite answer.

Since we can't fully answer statements 5 - 11 without more information about the second figure, we focus on the first - part answers:

Answer:

1. Lines parallel to $\overleftrightarrow{CG}$: $\overleftrightarrow{BF}$, $\overleftrightarrow{AE}$, $\overleftrightarrow{DH}$
2. Plane parallel to plane $DCGH$: plane $ABFE$
3. Skew lines to $\overleftrightarrow{EH}$: $\overleftrightarrow{AB}$, $\overleftrightarrow{BC}$, $\overleftrightarrow{CD}$, $\overleftrightarrow{DA}$, $\overleftrightarrow{BF}$, $\overleftrightarrow{AG}$
4. Lines parallel to plane $EFGH$: $\overleftrightarrow{AB}$, $\overleftrightarrow{BC}$, $\overleftrightarrow{CD}$, $\overleftrightarrow{DA}$