Linear Pair of Angles Worksheets - Free Printable

To solve the problem, we need to identify which of the given figures can be drawn as a single continuous line without lifting the pen or retracing any part of the line. This is a classic problem related to Eulerian paths in graph theory.

Key Concepts:

1. Graph Representation: Each figure can be represented as a graph where:
- Vertices are the points where lines meet.
- Edges are the lines connecting the vertices.

2. Degree of a Vertex: The degree of a vertex is the number of edges connected to it.

3. Eulerian Path:
- A graph has an Eulerian path if and only if it has exactly 0 or 2 vertices of odd degree.
- If there are 0 vertices of odd degree, the path is a closed loop (Eulerian circuit).
- If there are 2 vertices of odd degree, the path starts at one of these vertices and ends at the other.

Step-by-Step Solution:

#### Figure 1:
- Vertices: 4
- Degrees: All vertices have degree 2.
- Odd-degree vertices: 0
- Conclusion: This figure has an Eulerian circuit (can be drawn in one stroke).

#### Figure 2:
- Vertices: 5
- Degrees: One vertex has degree 3, and the rest have degree 2.
- Odd-degree vertices: 1
- Conclusion: This figure does not have an Eulerian path because it has more than 2 odd-degree vertices.

#### Figure 3:
- Vertices: 5
- Degrees: Two vertices have degree 3, and the rest have degree 2.
- Odd-degree vertices: 2
- Conclusion: This figure has an Eulerian path (can be drawn in one stroke).

#### Figure 4:
- Vertices: 6
- Degrees: Four vertices have degree 3, and two have degree 2.
- Odd-degree vertices: 4
- Conclusion: This figure does not have an Eulerian path because it has more than 2 odd-degree vertices.

#### Figure 5:
- Vertices: 5
- Degrees: One vertex has degree 3, and the rest have degree 2.
- Odd-degree vertices: 1
- Conclusion: This figure does not have an Eulerian path because it has more than 2 odd-degree vertices.

#### Figure 6:
- Vertices: 5
- Degrees: Two vertices have degree 3, and the rest have degree 2.
- Odd-degree vertices: 2
- Conclusion: This figure has an Eulerian path (can be drawn in one stroke).

Final Answer:

The figures that can be drawn as a single continuous line are:
- Figure 1
- Figure 3
- Figure 6

$$
\boxed{1, 3, 6}
$$