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Lines parallel to the same line
If two lines are parallel to the same line, we should prove that the two lines are also parallel to each other.

Let us look at the figure required to prove the statement given above.

Let the line $$AB$$ be parallel to line $$CD$$ and let line $$AB$$ also be parallel to $$EF$$.

We should prove that lines $$CD$$ and $$EF$$ are parallel.

$$\angle 1 = \angle 2 \longrightarrow (1)$$

[By corresponding angles axiom]

Similarly, $$\angle 1 = \angle 3 \longrightarrow (2)$$

On equating $$(1)$$ and $$(2)$$, we get:

$$\angle 2 = \angle 3$$

The angles $$\angle 2 = \angle 3$$ are corresponding angles.

Thus by the converse of corresponding angles axiom, we can prove that $$CD$$ is parallel to $$EF$$.

Theorem $$6$$: Lines which are parallel to the same line are parallel to each other.

Let us imagine that $$AB \parallel CD$$, and $$AB \parallel EF$$. $$PQ$$ intersects $$AB$$, $$CD$$, and $$EF$$.

We should prove that $$CD$$ $$\parallel EF$$.

Let us first consider the condition $$AB \parallel CD$$.

Here, $$\angle 1$$ is the corresponding angle of $$AB$$, and $$\angle 2$$ is the corresponding angle of $$CD$$.

Since $$AB \parallel CD$$, their corresponding angles are equal.

That is, $$\angle 1 = \angle 2 \longrightarrow (1)$$

Let us now consider the condition $$AB \parallel EF$$.

Even here, since $$AB$$ and $$EF$$ are parallel to each other, their corresponding angles are equal.

$$\angle 1$$ is the corresponding angle of $$AB$$, $$\angle 3$$ is the corresponding angle of $$EF$$.

Thus, $$\angle 1 = \angle 3 \longrightarrow (2)$$

On equating $$(1)$$ and $$(2)$$, we get:

$$\angle 2 = \angle 3$$

Here, $$\angle 2$$ is the corresponding angle of $$CD$$, and $$\angle 3$$ is the corresponding angle of $$EF$$.

Since the corresponding angles of $$CD$$ and $$EF$$ are equal, $$CD \parallel EF$$.