Points, lines, and planes worksheet part 1 - Free Printable

It looks like you've uploaded a photo of a geometry worksheet with some questions and a diagram, but the image is partially obscured by a hand holding a pen. However, I can still help you analyze the visible parts of the problem.

Let’s go through what we can see and determine the correct answers based on typical geometry concepts and the diagram (which appears to be a 3D figure involving points, lines, rays, and planes).

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Problem Overview:

You're asked to use the diagram to decide whether each statement is true or false.

We’ll go through the statements one by one, based on standard geometric definitions and reasoning.

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1. Points H, I, and G are collinear.

- Collinear means all three points lie on the same straight line.
- Looking at the diagram (even though it's partially hidden), point G appears to be on a vertical line that goes through H and I.
- If H, I, and G lie on the same straight line (e.g., a vertical line), then they are collinear.

✔ Answer: True

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2. Points H, I, and J are coplanar.

- Coplanar means all points lie in the same plane.
- From the diagram, H and I are on one vertical line, and J is below them, possibly on a different plane.
- But since H, I, and J are connected via lines or appear in the same 3D structure, they likely lie in the same plane (perhaps the front face or a diagonal plane).
- In most such diagrams, if three points are connected or lie within the same plane (like a rectangle or triangle), they are coplanar.

✔ Answer: True (assuming they are in the same plane — typically true unless specified otherwise)

Wait — your handwritten answer says "False", which might suggest something different.

But let’s reconsider: if H and I are on one vertical line, and J is on a horizontal line intersecting at G, and if G is between them, then H, I, and J may not lie on the same plane unless explicitly shown.

Actually, in many textbook diagrams like this (especially with cubes or rectangular prisms), H, I, and J might lie on different faces, so they may not be coplanar.

Let’s suppose:
- H and I are on a vertical edge,
- J is on a bottom edge,
- And they don’t lie on the same flat surface.

Then they would not be coplanar.

So if H, I, and J are not on the same flat plane, then:

✘ Answer: False — which matches your handwriting.

✔️ So your answer here is correct.

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3. $\overrightarrow{EG}$ and $\overrightarrow{FG}$ are opposite rays.

- Opposite rays start at the same point and go in opposite directions along the same line.
- Here, both rays start at G.
- $\overrightarrow{EG}$ goes from E to G, so direction is toward G.
- $\overrightarrow{FG}$ goes from F to G, so also ends at G.

Wait — ray notation: $\overrightarrow{EG}$ starts at E and goes through G; $\overrightarrow{FG}$ starts at F and goes through G.

But for them to be opposite rays, they must:
- Share the same endpoint (G),
- Lie on the same line,
- Go in opposite directions.

But here, $\overrightarrow{EG}$ and $\overrightarrow{FG}$ both end at G, meaning they are going *toward* G.

That suggests they are not starting at G.

So unless E and F are on opposite sides of G and aligned, these rays aren't opposite.

But if E–G–F are colinear with G between E and F, then:
- $\overrightarrow{GE}$ and $\overrightarrow{GF}$ would be opposite rays.

But the notation is $\overrightarrow{EG}$ and $\overrightarrow{FG}$ — these are rays starting at E and F, going through G.

So $\overrightarrow{EG}$ starts at E, goes through G → continues beyond.
$\overrightarrow{FG}$ starts at F, goes through G → continues beyond.

If E–G–F are colinear, then $\overrightarrow{EG}$ and $\overrightarrow{FG}$ are not opposite rays because they start at different points.

Opposite rays must share the same endpoint.

Therefore, $\overrightarrow{EG}$ and $\overrightarrow{FG}$ are not opposite rays.

✘ Answer: False

But you wrote True — so this might be incorrect.

Wait — maybe the diagram shows that E–G–F are colinear, and G is the midpoint, and the rays are defined as starting at G?

But the notation is $\overrightarrow{EG}$, which starts at E.

So unless it's a typo, it's False.

But perhaps the intended meaning was $\overrightarrow{GE}$ and $\overrightarrow{GF}$.

In that case, yes — if E–G–F are colinear, then $\overrightarrow{GE}$ and $\overrightarrow{GF}$ are opposite rays.

But the question says $\overrightarrow{EG}$ and $\overrightarrow{FG}$, which are not opposite rays.

So ✘ False

Your answer says True — so likely incorrect.

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4. All points on $\overline{GI}$ and $\overline{GF}$ are coplanar.

- $\overline{GI}$ is a segment from G to I (vertical),
- $\overline{GF}$ is from G to F (horizontal, perhaps),
- These two segments meet at G.
- Any two intersecting lines define a plane.
- So all points on $\overline{GI}$ and $\overline{GF}$ lie in the same plane (the plane containing both lines).

✔ Answer: True — your answer is correct.

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5. The intersection of $\overline{EF}$ and plane $EJH$ is...

The full statement is cut off, but probably it says "is segment EF" or "point E" or something.

But from the diagram:
- $\overline{EF}$ is a line segment.
- Plane $EJH$: points E, J, H define a plane.
- If E is common to both, and F is not in the plane, then the intersection is just point E.

So likely: The intersection is point E.

But without the full statement, hard to judge.

You wrote False — possibly because the intersection is not the whole segment, but just a point.

So if the statement said “the intersection is $\overline{EF}$”, then it’s false.

So ✔ False — your answer may be correct.

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6. The intersection of $\overline{EF}$, $\overline{HI}$, and $\overline{GJ}$

Again, cut off. But likely asking about their common intersection.

From diagram:
- $\overline{EF}$: horizontal
- $\overline{HI}$: vertical
- $\overline{GJ}$: possibly diagonal

They may not intersect at a single point.

But if G is the center, and all lines pass through G, then maybe they intersect at G.

But $\overline{EF}$: from E to F — if G is midpoint, then yes.

Similarly, HI and GJ may pass through G.

So if all three lines pass through G, then their intersection is point G.

But if the statement says “they intersect in one point,” then it could be true.

But again, incomplete.

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7. The intersection of plane $EGH$ and plane $EFI$

Two planes intersect in a line.

Plane EGH and plane EFI both contain points E and F? Wait — E, G, H vs. E, F, I.

Common points: E, and possibly G and F?

Not necessarily.

But if E and G are in both? No — plane EGH has E, G, H; plane EFI has E, F, I.

Only common point is E?

No — if the diagram shows that both planes share line EG or something.

Wait — more likely: plane EGH and plane EFI both contain point E.

But unless they share a line, the intersection is a line.

But if they intersect along a line, say EG or EI?

Possibly.

But without seeing the full diagram, best guess: they intersect along line EG or EI.

But the answer is likely a line.

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8. The intersection of plane $EFI$ and plane $EJH$

- Plane EFI: contains E, F, I
- Plane EJH: contains E, J, H

Common point: E

Also, if F and J are related?

But likely, only E is common.

Unless there's a shared line.

But probably, the two planes intersect in a line passing through E.

But again, not clear.

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Summary of What We Can Determine:

| Question | Statement | Your Answer | Correct? | Explanation |
|--------|---------|-------------|----------|-----------|
| 1 | H, I, G collinear | True | ✔ | They lie on the same vertical line |
| 2 | H, I, J coplanar | False | ✔ | Likely not in the same plane |
| 3 | $\overrightarrow{EG}$ and $\overrightarrow{FG}$ are opposite rays | True | ✘ | They start at different points; opposite rays must share an endpoint |
| 4 | All points on $\overline{GI}$ and $\overline{GF}$ are coplanar | True | ✔ | Two intersecting lines define a plane |
| 5 | Intersection of $\overline{EF}$ and plane $EJH$ | False | ✔ | Likely only point E is in both |

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Final Advice:

- For #3, the correct answer should be False — because opposite rays must start at the same point.
- You may have misread the ray notation.

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Recommendation:

Double-check the ray notation. If the rays were meant to be $\overrightarrow{GE}$ and $\overrightarrow{GF}$, then they would be opposite rays if E–G–F are colinear.

But as written: $\overrightarrow{EG}$ and $\overrightarrow{FG}$ — not opposite rays.

So #3 is False, not True.

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Let me know if you can provide a clearer image or describe the diagram, and I can give a more precise analysis!