Skills Practice worksheet on Points, Lines, and Planes from Glencoe Geometry, including diagrams and questions for identifying geometric elements.

Let's solve each problem step by step based on the given figures and geometric concepts.

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Figure 1 (Top Figure):

This figure shows a plane with points, lines, and labels:

- Points: A, B, C, D, E
- Lines: $ \ell $, $ m $, $ p $
- Plane: Shown as a parallelogram (often labeled as a flat surface)

We’ll use this to answer questions 1–4.

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#### 1. Name a line that contains point D.

Look at the figure: Point D lies on line $ p $.

✔ Answer: Line $ p $

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#### 2. Name a point contained in line $ \pi $.

Wait — there is no line labeled $ \pi $ in the figure. But we see lines labeled $ \ell $, $ m $, and $ p $. Possibly a typo? Looking again — it says "line $ \pi $" but likely meant line $ m $ or line $ \ell $?

But wait — in the diagram, line $ m $ passes through points A and D. And point A lies on line $ m $.

Wait — let’s double-check.

Actually, looking closely:

- Line $ \ell $ goes from B to C.
- Line $ m $ goes from A to D.
- Line $ p $ goes from D to E.

So if the question asks for a point on line $ \pi $, but there's no $ \pi $, perhaps it's a typo and should be line $ m $ or line $ p $?

But the question says: "Name a point contained in line $ \pi $" — but $ \pi $ is not in the figure.

Wait — maybe $ \pi $ is meant to be $ m $? Or could it be a labeling issue?

Alternatively, in some textbooks, Greek letters like $ \pi $ are used for planes, not lines.

Ah! That might be it. Let's reconsider.

In geometry:
- Lines are usually denoted by lowercase letters: $ \ell $, $ m $, $ n $, etc.
- Planes are often denoted by uppercase script letters or Greek letters like $ \pi $.

But in this figure, no plane is labeled $ \pi $. The plane is just shown as a shaded quadrilateral.

So this may be a typo.

Wait — actually, in question 4, it says: “Name the plane containing lines $ \pi $ and $ p $” — so now we know $ \pi $ is likely a line, not a plane.

But there is no line labeled $ \pi $ in the diagram.

Wait — perhaps $ \pi $ is meant to be $ m $? Because line $ m $ contains point D and runs from A to D.

But then why call it $ \pi $? That doesn’t make sense.

Alternatively, could $ \pi $ be a typo for $ \ell $?

Let’s look again.

Wait — perhaps the figure has a line labeled $ \pi $? Let’s re-express the image description.

From the image:
- There is a plane (parallelogram).
- Points: A, B, C, D, E
- Lines: $ \ell $, $ m $, $ p $
- $ \ell $: connects B to C
- $ m $: connects A to D
- $ p $: connects D to E

So only lines $ \ell $, $ m $, $ p $ are labeled.

Therefore, "line $ \pi $" is likely a typo.

Possibility: Maybe $ \pi $ was meant to be $ m $?

But in question 3: “What is another name for line $ p $?” — so $ p $ is a line.

Then in question 2: “Name a point contained in line $ \pi $” — unless $ \pi $ is not a line, but a plane?

Wait — in question 4: “Name the plane containing lines $ \pi $ and $ p $” — so $ \pi $ must be a line.

But there is no such line in the diagram.

Wait — unless $ \pi $ is meant to be $ m $? Because both $ m $ and $ p $ intersect at D.

But $ m $ is already labeled.

Wait — perhaps the label $ \pi $ is missing from the diagram?

No — in the original image, the line from A to D is labeled $ m $, B to C is $ \ell $, D to E is $ p $.

So unless there’s an error in the textbook, this seems like a typo.

But let’s assume that $ \pi $ is a typo and should be $ m $ — because $ m $ is a line that contains point A and D.

Alternatively, maybe $ \pi $ refers to line $ \ell $?

But line $ \ell $ goes from B to C.

Points on $ \ell $: B and C.

So if $ \pi $ = $ \ell $, then points B and C lie on it.

But without knowing what $ \pi $ is, we can't proceed.

Wait — perhaps $ \pi $ is not a line label, but a plane? But then it would be inconsistent with “line $ \pi $”.

Another possibility: In some texts, $ \pi $ is used for a plane, and $ \ell $ for a line.

But here, the question says: “line $ \pi $”, which suggests $ \pi $ is a line.

Given the confusion, and since only lines $ \ell $, $ m $, $ p $ are labeled, I suspect “$ \pi $” is a typo and should be $ m $.

So:

> Question 2: Name a point contained in line $ m $.

Then: Points A and D lie on line $ m $.

✔ Answer: Point A (or D)

But let’s move to question 3, which is clearer.

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#### 3. What is another name for line $ p $?

Line $ p $ goes from D to E.

So, any two points on the line can name it.

Points on line $ p $: D and E.

So another name for line $ p $ is line DE or line ED.

✔ Answer: Line DE

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#### 4. Name the plane containing lines $ \pi $ and $ p $.

Again, $ \pi $ is undefined.

But if $ \pi $ is meant to be $ m $, then lines $ m $ and $ p $ both lie in the same plane.

They intersect at point D.

So the plane containing both lines $ m $ and $ p $ is the entire plane shown in the figure.

But how is the plane labeled?

It’s not labeled with a symbol. But in the second figure (bottom), the plane is labeled $ \mathcal{Q} $.

But in this top figure, the plane isn't labeled.

So probably, we just say: the plane shown in the figure.

But typically, we name a plane using three non-collinear points.

So pick three non-collinear points in the plane.

For example: A, B, C — all in the plane.

Or A, C, D.

But since no label is given, we can say:

✔ Answer: The plane containing points A, B, C, D, E (or simply "the plane shown")

But better: Since lines $ m $ and $ p $ intersect at D, and lie in the plane, and the whole figure is one plane, we can say:

✔ Answer: The plane determined by points A, D, and E (or any three non-collinear points)

But without a label, we can’t give a symbolic name.

Wait — in the second figure, the plane is labeled $ \mathcal{Q} $, but in the first figure, it's not.

So likely, the answer is: The plane containing the figure, or the plane formed by points A, B, C, D, E.

But let’s assume that the plane is unnamed, so we describe it.

✔ Answer: The plane containing points A, B, C, D, and E

But since the question asks to "name" it, and it's not labeled, perhaps we just say: Plane ABC or Plane ADE, etc.

But best answer: The plane containing lines $ m $ and $ p $ — but that's circular.

Alternatively, since lines $ m $ and $ p $ intersect at D, and both lie in the plane, the plane is uniquely determined.

But unless labeled, we can't give a name.

Wait — maybe the plane is labeled in the figure? Let's check.

Looking back: The top figure has a parallelogram with points A, B, C, D, E. No plane label.

So likely, the answer is: the plane containing points A, B, C, D, and E.

But let’s skip for now and come back.

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Now, questions 5–8: Draw and label a figure for each relationship.

These are drawing tasks, but we can describe them.

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#### 5. Point K lies on $ \overline{RT} $.

- Draw a segment $ \overline{RT} $.
- Place point K somewhere on the segment between R and T.

✔ Drawing description: Segment RT with point K on it.

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#### 6. Plane $ f $ contains line $ s $.

- Draw a plane (a flat surface), label it $ f $.
- Draw a line $ s $ lying entirely within the plane.

✔ Drawing description: A plane labeled $ f $, with a line $ s $ drawn inside it.

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#### 7. $ \overline{YP} $ lies in plane $ \mathcal{Q} $ and contains point C, but does not contain point H.

- Draw a plane $ \mathcal{Q} $.
- Draw segment $ \overline{YP} $ inside the plane.
- Place point C on $ \overline{YP} $.
- Place point H outside of $ \overline{YP} $, but possibly in the plane or not — but since it says "does not contain point H", H is not on the segment.

So H can be in the plane but not on the segment.

✔ Drawing description: Plane $ \mathcal{Q} $, segment $ \overline{YP} $ in it, point C on $ \overline{YP} $, point H in the plane but not on the segment.

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#### 8. Lines $ q $ and $ f $ intersect at point Z in plane $ \mathcal{L} $.

- Draw a plane $ \mathcal{L} $.
- Draw two lines $ q $ and $ f $ that cross at point Z.
- Both lines lie in plane $ \mathcal{L} $.

✔ Drawing description: Plane $ \mathcal{L} $, two lines $ q $ and $ f $ crossing at Z, both in the plane.

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Now, refer to the second figure (bottom one):

This is a 3D figure showing a rectangular prism-like shape.

Points: A, B, C, D, E, F, G, H

Labeling:
- Bottom face: A, B, C, D
- Top face: E, F, G, H
- Edges: AE, BF, CG, DH

And a plane $ \mathcal{Q} $ is labeled, which appears to be the front face: A, B, F, E.

Also, lines $ g $ and $ f $ are mentioned in Q8, but not in this figure.

But questions 9–12 refer to this figure.

Let’s analyze.

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#### 9. How many planes are shown in the figure?

We see:
- Bottom face: ABCD
- Top face: EFGH
- Front face: ABFE
- Back face: CDHG
- Left face: ADHE
- Right face: BCGF

That’s 6 faces — each is a plane.

Additionally, plane $ \mathcal{Q} $ is labeled — it's the front face (ABFE).

But the figure shows these 6 planes.

Are they all shown? Yes — the 3D box has 6 faces.

✔ Answer: 6 planes

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#### 10. How many of the planes contain points F and E?

Points F and E are connected by edge EF.

Which planes contain both F and E?

- Top face: EFGH → contains E and F ✔
- Front face: ABFE → contains E and F ✔
- Left face: ADHE → contains E, but not F ✘
- Right face: BCGF → contains F, but not E ✘
- Bottom: ABCD → no E or F ✘
- Back: CDHG → no E or F ✘

So only two planes contain both E and F:
- Top face: EFGH
- Front face: ABFE

✔ Answer: 2 planes

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#### 11. Name four points that are coplanar.

Coplanar = lying in the same plane.

Any four points on one face.

For example:
- A, B, F, E — all on front face
- A, B, C, D — bottom face
- E, F, G, H — top face

✔ Answer: A, B, F, E (or any four points on one face)

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#### 12. Are points A, B, and C coplanar? Explain.

Yes. Points A, B, C are on the bottom face ABCD, which is a plane.

Any three points define a plane, but here they lie on a known plane.

Since A, B, C are on the bottom face, and the bottom face is planar, they are coplanar.

Even if you take just three points, they always determine a plane — so yes.

But the question is whether they are coplanar — meaning do they lie in a single plane?

Yes, they lie in the plane of the bottom face.

✔ Answer: Yes, points A, B, and C are coplanar because they all lie in the same plane (the bottom face ABCD).

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Revisit Questions 1–4 with Clarification

Now that we’ve done the rest, let’s go back.

Looking at the top figure, perhaps $ \pi $ is a typo and should be $ m $?

Because line $ m $ goes from A to D.

And line $ p $ goes from D to E.

So:

#### 1. Name a line that contains point D.

Point D is on lines $ m $ and $ p $.

So answers: line $ m $ or line $ p $

✔ Answer: Line $ m $ or line $ p $

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#### 2. Name a point contained in line $ \pi $.

If $ \pi $ is a typo for $ m $, then points A and D are on line $ m $.

✔ Answer: Point A or D

But if $ \pi $ is meant to be $ \ell $, then B and C are on it.

But since $ \ell $ is labeled, and $ \pi $ is not, likely a typo.

Perhaps $ \pi $ is meant to be line $ \ell $?

But $ \ell $ is already labeled.

Alternatively, maybe $ \pi $ is a plane?

But the question says “line $ \pi $”.

Wait — in some textbooks, $ \pi $ is used for a plane.

But here, it says “line $ \pi $”, so it must be a line.

Given the ambiguity, and since only $ \ell $, $ m $, $ p $ are labeled, I think $ \pi $ is a typo.

Assuming $ \pi $ = $ m $, then:

✔ Answer to 2: Point A (or D)

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#### 3. What is another name for line $ p $?

Line $ p $ goes from D to E.

So: line DE or line ED

✔ Answer: Line DE

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#### 4. Name the plane containing lines $ \pi $ and $ p $.

If $ \pi $ = $ m $, then lines $ m $ and $ p $ intersect at D.

They both lie in the same plane — the plane of the figure.

We can name the plane using three non-collinear points.

For example: A, B, C — all in the plane.

Or A, D, E.

So: Plane ABC, Plane ADE, or just the plane shown

But since no label is given, we can say:

✔ Answer: The plane containing points A, B, C, D, and E

Or more precisely: Plane ABC (since A, B, C are not collinear and define the plane)

But the safest answer is: The plane containing the figure

But if we need a specific name, we can say: Plane ABC

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✔ Final Answers:

1. Line $ m $ or line $ p $
2. Point A (assuming $ \pi $ = $ m $)
3. Line DE
4. The plane containing points A, B, C, D, and E (or Plane ABC)
5. Draw segment $ \overline{RT} $ with point K on it.
6. Draw plane $ f $ with line $ s $ lying in it.
7. Draw plane $ \mathcal{Q} $, segment $ \overline{YP} $ in it, point C on it, point H not on it.
8. Draw plane $ \mathcal{L} $, lines $ q $ and $ f $ intersecting at Z, both in the plane.
9. 6 planes
10. 2 planes
11. A, B, F, E (or any four points on one face)
12. Yes, because points A, B, and C lie on the same plane (the bottom face ABCD).

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Let me know if you'd like diagrams sketched or further explanation!