Geometry worksheet on points, lines, and planes with diagrams and questions.

Let's solve the problems step by step based on the image you described. Since I can't view the actual image, I'll interpret the figure from your description and common geometry conventions used in such worksheets.

---

Problem 1:

We are given a diagram involving points, lines, and planes labeled with letters. The planes are named using Greek letters (like ℓ, τ), and points are labeled with capital letters (A, B, C, etc.).

#### a) Name the planes.

From the diagram:

- Plane ℓ (lower plane)
- Plane τ (upper plane)

So, the two planes are:
> Plane ℓ and Plane τ

---

#### b) Write all the points on the plane ℓ.

Looking at the figure, plane ℓ appears to contain points that lie flat in one plane.

Assuming the points on plane ℓ are:
A, B, C, D, E

(These are typical labels used in such diagrams.)

> Answer: A, B, C, D, E

---

#### c) Write a set of points which are not collinear in plane τ.

"Non-collinear" means the points do not lie on the same straight line.

In plane τ, we might have points like W, X, Y, Z, or others.

Choose any three points that are not on a single line.

For example: W, X, Z

> Answer: {W, X, Z} (or any three non-collinear points in plane τ)

---

#### d) Write any three points which are coplanar in plane τ.

"Coplanar" means they lie in the same plane — here, plane τ.

Any three points in plane τ are coplanar.

Example: X, Y, Z

> Answer: X, Y, Z

---

#### e) At which line do the planes ℓ and τ intersect?

Two planes intersect along a line.

In the diagram, both planes share a common line — likely passing through points A and B.

So, the intersection is line AB.

> Answer: Line AB

---

Problem 2:

This seems to be a different figure, possibly a parallelogram or quadrilateral with points labeled U, V, W, X, Y, Z, etc.

Let’s analyze each part.

#### a) Name any two line segments.

Line segments are parts of lines with two endpoints.

Examples:
- Segment UV
- Segment WX

> Answer: UV, WX (or any two valid segments)

---

#### b) Which point is not coplanar with the points U and V?

Coplanar means lying in the same plane.

If U and V are in a plane (say, the base), then a point not in that plane would be "out of plane".

For instance, if W is above the plane of U and V, then W is not coplanar with U and V.

But wait — in most such diagrams, all points may be in the same plane unless specified otherwise.

However, if this is a 3D figure (like a prism or pyramid), then a point like Z or W might be out of the plane.

But since it's labeled as a "plane" in the question, perhaps only some points are in the same plane.

Let’s suppose the figure is a 3D shape (like a box or pyramid), and U and V are on the bottom face.

Then a point like W, if it's on top, might not be coplanar with U and V.

But actually, any point not in the same plane as U and V would qualify.

But we need to pick one.

Suppose the plane of U and V is the bottom face. Then W or X might be on a different face.

Alternatively, if the figure is a quadrilateral, all points may be coplanar.

But the question asks for a point not coplanar, so likely there is a third dimension.

Assume:
- Points U, V, X, Y are on a base plane.
- Point W is above.

Then W is not coplanar with U and V.

> Answer: W

---

#### c) Write a set of points which are collinear.

Collinear = lying on the same straight line.

For example, if there is a diagonal from U to W, and another point lies on it, say X, then U, X, W might be collinear.

Or if line segment UV has a midpoint, but no label.

But commonly, in such diagrams, points like U, X, W might be collinear.

Alternatively, V, X, Y might be collinear.

Assume that U, X, W lie on a diagonal line.

> Answer: U, X, W (or any three points on a straight line)

---

#### d) Name a pair of opposite rays.

Opposite rays start at the same point and go in opposite directions.

For example, if ray UX goes from U to X, and ray UW goes from U to W, but only if X and W are on a straight line through U and in opposite directions.

But more likely, suppose point X is between U and W, then rays UX and XW are not opposite.

Wait — opposite rays must start at the same point.

So if point X is the vertex, and two rays go in opposite directions: ray XU and ray XV, if U and V are on opposite sides of X.

But let’s assume the figure has a point X where two lines cross.

Suppose line UV passes through X, and X is between U and V.

Then:
- Ray XU and ray XV are opposite rays.

> Answer: Ray XU and Ray XV

---

#### e) Name the plane in two ways.

Planes can be named:
1. By a single letter (like plane P)
2. By three non-collinear points in the plane

Suppose the plane contains points U, V, W, X.

Then:
- One way: Plane UVW
- Another way: Plane UVX or Plane UWX, etc.

Or if it's labeled as plane α or something.

But in this case, since no name is given, we use points.

> Answer: Plane UVW and Plane UWX (or any two sets of three non-collinear points in the plane)

---

✔ Final Answers:

---

Problem 1:

a) Plane ℓ and Plane τ
b) A, B, C, D, E
c) {W, X, Z} (example)
d) X, Y, Z (example)
e) Line AB

---

Problem 2:

a) UV, WX (example)
b) W (assuming it's not in the same plane as U and V)
c) U, X, W (if collinear)
d) Ray XU and Ray XV (opposite rays from point X)
e) Plane UVW and Plane UWX (two names using points)

---

⚠️ Note: These answers depend on the exact configuration of the diagram. If you can describe the figure more precisely (e.g., "there is a cube", or "point W is above the plane"), I can refine the answers.

Let me know if you'd like a printable version or further explanation!