Geometry Skills Practice - Free Printable

Let's solve each problem step by step based on the provided image and geometric principles.

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Figure Reference (Top Right):

We see a parallelogram-like figure with labeled points:
- Points: A, B, C, D, E, F, G
- Lines: Line $ p $, line $ n $
- Planes: Not explicitly named but implied

Let’s go through each question:

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1. Name a line that contains point D.

From the diagram, point D lies on line $ p $.
Answer: Line $ p $

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2. Name a point contained in line $ n $.

Line $ n $ passes through points C and E. So any of these points are on line $ n $.
Answer: Point C or Point E

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

A line can be named using any two points on it. Line $ p $ goes through points B and D.
So, another name is line BD (or DB).
Answer: Line BD

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4. Name the plane containing lines $ n $ and $ p $.

Lines $ n $ and $ p $ intersect at point C (they appear to meet at C), and both lie in the same flat surface — the quadrilateral ABCD. This suggests they lie in plane ABCD, which is the entire figure.

But since the plane isn't labeled, we use three non-collinear points to define it. Points A, B, and C are not collinear and lie in this plane.

Alternatively, since the figure looks like a parallelogram, and all points lie in one plane, we can call it plane ABC or plane ABCD.

Answer: Plane ABCD (or Plane ABC)

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5. Draw and label a figure for: Point K lies on $ \overline{RT} $.

This means point K is somewhere on the segment from R to T.

Drawing Instructions:
- Draw a line segment RT.
- Mark a point K anywhere between R and T (or possibly at an endpoint).
- Label the points R, T, and K.

✔ Example:
```
R ---- K ---- T
```

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6. Draw and label a figure for: Plane $ j $ contains line $ s $.

- Draw a plane (a flat surface, often represented as a parallelogram).
- Draw a line $ s $ lying entirely within that plane.
- Label the plane $ j $ and the line $ s $.

✔ Example:
```
Plane j
___________
| s |
| ----- |
|__________|
```

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7. Draw and label a figure for: $ \overline{YP} $ lies in plane $ \mathscr{B} $ and contains point C, but does not contain point H.

- Draw a plane $ \mathscr{B} $ (label it).
- Draw segment $ \overline{YP} $ inside the plane.
- Place point C on segment $ \overline{YP} $.
- Place point H outside the segment $ \overline{YP} $, but still in the plane (or even better, show H not on $ \overline{YP} $).

✔ Example:
```
Plane B
+------------------+
| Y----C----P |
| H |
+------------------+
```
Note: H is in the plane but not on $ \overline{YP} $

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8. Draw and label a figure for: Lines $ q $ and $ f $ intersect at point Z in plane $ \mathscr{U} $.

- Draw a plane $ \mathscr{U} $.
- Draw two lines $ q $ and $ f $ crossing at point Z.
- Label everything.

✔ Example:
```
Plane U
+------+
| q |
| / |
|/ |
Z |
| \ |
| \f |
|______|
```

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Refer to the second figure (bottom right):

This shows a 3D shape resembling a triangular prism or pyramid with labeled points:
- Points: A, B, C, D, E, F, G, H
- Planes: Not labeled, but implied by faces.

It appears to be a triangular prism with triangle ABC on the bottom and triangle DEF on top, connected by edges.

Wait — actually, looking closely:
- Bottom face: A, B, C
- Top face: D, E, F
- But also labeled G and H? Wait — labels are: A, B, C, D, E, F, G, H — but only six points visible?

Actually, the figure shows:

- Triangle ABC (base)
- Triangle DEF (top), with D above A, E above B, F above C?
- But there's also G and H? Wait — no, in the figure:

Looking carefully:
- Points: A, B, C, D, E, F, G, H
- It seems like a cube or rectangular prism? But labeled differently.

Wait — perhaps it's a rectangular box with:

- Base: A, B, C, D
- Top: E, F, G, H

But the diagram shows:

- A connected to B, B to C, C to D, D to A → base rectangle
- Then vertical edges: A to E, B to F, C to G, D to H
- Top: E, F, G, H

So it's a rectangular prism with:

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

Now let’s answer the questions.

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9. How many planes are shown in the figure?

In a rectangular prism, there are 6 faces, each a plane:

1. Bottom: ABCD
2. Top: EFGH
3. Front: ABFE
4. Back: CDHG
5. Left: ADHE
6. Right: BCGF

Each face is a plane.

Answer: 6 planes

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

Points E and F are connected by an edge (EF), and they lie on multiple faces.

Which planes contain both E and F?

- Top face: EFGH → contains E and F ✔
- Front face: ABFE → contains E and F ✔
- Side face: BCGF → contains F, but not E? No — wait:
- BCGF: B, C, G, F → includes F, but not E
- ABFE: A, B, F, E → yes ✔
- EFGH: E, F, G, H → yes ✔

So only two planes contain both E and F:

- Plane ABFE (front)
- Plane EFGH (top)

Answer: 2 planes

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11. Name four points that are coplanar.

Coplanar = lying in the same plane.

Any face has 4 points. For example:

- Points A, B, F, E → lie on front face ABFE
- Or A, B, C, D → lie on bottom face
- Or E, F, G, H → top face

Answer: A, B, F, E (or any set of 4 points on one face)

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

Yes. Points A, B, and C lie on the bottom face (ABCD), which is a plane.

Three points always define a plane unless they're colinear. Here, A, B, C form a triangle, so they are not colinear, and thus determine a unique plane.

Since they lie on the same face (rectangle ABCD), they are definitely coplanar.

Answer: Yes, because points A, B, and C lie on the same plane (the bottom face of the prism).

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✔ Final Answers:

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

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