2. points h, i, and j are coplanar.
3. $overrightarrow{eg}$ and $overrightarrow{fg}$ are opposite rays.
4. all points on $overrightarrow{gi}$ and $overrightarrow{gf}$ are coplanar.
5. the intersection of $overrightarrow{ef}$ and plane jkh is $overrightarrow{hi}$.
6. the intersection of $overrightarrow{ef}$, $overrightarrow{hi}$, and $overrightarrow{jg}$ is point g.
7. the intersection of plane egh and plane jgi is point g.
8. the intersection of plane efi and plane jkg is $overrightarrow{hg}$.
sketch the figure described.
9. two rays that do not intersect
10. three planes that intersect in one line
11. three lines that intersect in three points
12. a ray that intersects a plane in one point

Question

Accepted Answer

### Question 2: Points $ H $, $ I $, and $ J $ are coplanar.
# Answer:
True
# Explanation:
## Step1: Recall coplanar definition
Coplanar points lie on the same plane.
## Step2: Analyze the figure
From the diagram, $ H $, $ I $, $ J $ appear to lie on the vertical plane (or the plane containing $ J $, $ I $, $ H $ as per the sketch), so they are coplanar.

### Question 3: $ \overrightarrow{EG} $ and $ \overrightarrow{FG} $ are opposite rays.
# Answer:
True
# Explanation:
## Step1: Recall opposite rays definition
Opposite rays share the same endpoint and form a straight line (180° angle).
## Step2: Analyze the rays
$ \overrightarrow{EG} $ and $ \overrightarrow{FG} $ share endpoint $ G $ and lie on the straight line $ EF $ (opposite directions), so they are opposite rays.

### Question 4: All points on $ \overrightarrow{GI} $ and $ \overrightarrow{GF} $ are coplanar.
# Answer:
True
# Explanation:
## Step1: Recall coplanar lines
If two rays (or lines) intersect, all their points lie on the same plane (the plane formed by them).
## Step2: Analyze the rays
$ \overrightarrow{GI} $ and $ \overrightarrow{GF} $ intersect at $ G $, so all their points lie on the plane containing both rays.

### Question 5: The intersection of $ \overleftrightarrow{EF} $ and plane $ JKH $ is $ \overleftrightarrow{HI} $.
# Answer:
False
# Explanation:
## Step1: Recall line - plane intersection
A line and a plane intersect at a point (if the line is not in the plane) or the line itself (if the line is in the plane).
## Step2: Analyze $ \overleftrightarrow{EF} $ and plane $ JKH $
$ \overleftrightarrow{EF} $ is a horizontal line (from the diagram) and plane $ JKH $ is vertical. They intersect at point $ G $ (or a single point), not the line $ \overleftrightarrow{HI} $.

### Question 6: The intersection of $ \overleftrightarrow{EF} $, $ \overleftrightarrow{HI} $, and $ \overleftrightarrow{JG} $ is point $ G $.
# Answer:
True
# Explanation:
## Step1: Recall intersection of multiple lines
The intersection of three lines is a point that lies on all three lines.
## Step2: Analyze the lines
From the diagram, $ \overleftrightarrow{EF} $, $ \overleftrightarrow{HI} $, and $ \overleftrightarrow{JG} $ all pass through point $ G $, so their intersection is $ G $.

### Question 7: The intersection of plane $ EGH $ and plane $ JGI $ is point $ G $.
# Answer:
False
# Explanation:
## Step1: Recall plane - plane intersection
Two planes intersect in a line (if they are not parallel).
## Step2: Analyze the planes
Plane $ EGH $ and plane $ JGI $ intersect along the line $ \overleftrightarrow{JG} $ (or a line containing $ G $ and other points), not just the point $ G $.

### Question 8: The intersection of plane $ EFI $ and plane $ JKG $ is $ \overleftrightarrow{HG} $.
# Answer:
False
# Explanation:
## Step1: Recall plane - plane intersection
Two planes intersect in a line. We need to find the line common to both planes.
## Step2: Analyze the planes
Plane $ EFI $ and plane $ JKG $ do not intersect along $ \overleftrightarrow{HG} $. From the diagram, their intersection should be a line related to the vertical or horizontal lines, but $ \overleftrightarrow{HG} $ is not the line of intersection.

### Question 9: Sketch two rays that do not intersect
# Answer:
(Textual description for sketching)
Draw two rays in different planes or in the same plane but with non - intersecting directions. For example, draw ray $ \overrightarrow{AB} $ (starting at $ A $, going to the right) and ray $ \overrightarrow{CD} $ (starting at $ C $, going up) where $ A $, $ B $, $ C $, $ D $ are placed such that the rays do not meet. If in the same plane, draw ray $ \overrightarrow{EF} $ (along the x - axis, starting at $ E(0,0) $, going to $ F(1,0) $) and ray $ \overrightarrow{GH} $ (along the line $ y = 1 $, starting at $ G(2,1) $, going to $ H(3,1) $).
# Explanation:
## Step1: Recall ray properties
A ray has a starting point and extends infinitely in one direction.
## Step2: Determine non - intersecting condition
Two rays do not intersect if their paths (the lines containing them) are parallel or if they are in different planes, or if their starting points and directions are such that their infinite extensions do not meet.

### Question 10: Sketch three planes that intersect in one line
# Answer:
(Textual description for sketching)
Imagine three books (representing planes) stacked in a way that they all share a common spine (the line of intersection). For example, let the first plane be the $ xy $ - plane, the second plane be a plane that contains the $ z $ - axis and is at an angle to the $ xy $ - plane, and the third plane be another plane that also contains the $ z $ - axis. All three planes intersect along the $ z $ - axis.
# Explanation:
## Step1: Recall plane - plane intersection
Two planes intersect in a line. For three planes to intersect in one line, all three planes must contain that common line.
## Step2: Construct the sketch
We can use the idea of a common line (e.g., the $ z $ - axis in a 3D coordinate system) and define three planes that all pass through this line.

### Question 12: Sketch a ray that intersects a plane in one point
# Answer:
(Textual description for sketching)
Draw a plane (e.g., a square representing a plane) and a ray that starts outside the plane and enters the plane, intersecting it at one point. For example, let the plane be the $ xy $ - plane ( $ z = 0 $ ). Draw a ray starting at $ A(0,0,1) $ and going towards $ B(0,0, - 1) $. This ray intersects the $ xy $ - plane ( $ z = 0 $) at the point $ (0,0,0) $.
# Explanation:
## Step1: Recall ray - plane intersection
A ray (which is a part of a line) intersects a plane at a point if the ray is not parallel to the plane and does not lie entirely in the plane.
## Step2: Construct the sketch
Choose a ray with a starting point outside the plane and a direction that will pass through the plane, resulting in a single intersection point.

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QUESTION IMAGE

Question

Explanation:

Question 2: Points \( H \), \( I \), and \( J \) are coplanar.

Step1: Recall coplanar definition

Step2: Analyze the figure

Question 3: \( \overrightarrow{EG} \) and \( \overrightarrow{FG} \) are opposite rays.

Step1: Recall opposite rays definition

Step2: Analyze the rays

Question 4: All points on \( \overrightarrow{GI} \) and \( \overrightarrow{GF} \) are coplanar.

Step1: Recall coplanar lines

Step2: Analyze the rays

Question 5: The intersection of \( \overleftrightarrow{EF} \) and plane \( JKH \) is \( \overleftrightarrow{HI} \).

Answer: