We are given that $r\parallel s$ . Also, the transversal is shown. Let us take the first case, that of $\angle 1$ and $\angle 5$ . Please note that all other proofs will follow in a similar manner.

Let us begin, please have a nice look at the diagram. We will see that $\angle 1$ and $\angle 3$ are vertically opposite angles. We know that vertically opposite angles are congruent. Thus, $\angle 1$ and $\angle 3$ are congruent angles.

$\angle 1$ = $\angle 3$

Now, we know that $\angle 3$ and $\angle 5$ are alternate interior angles. We also, know that alternate interior angles are equal too. Thus, we have:

$\angle 3$ = $\angle 5$

From the above arguments it is clear that:

$\angle 1$ = $\angle 3$ = $\angle 3$ .

Thus, $\angle 1$ = $\angle 3$

We have proven the first instance. Please note that all other instances can be proved in a similar fashion.

For example, for $\angle 2$ and $\angle 6$ we can take $\angle 2$ and $\angle 4$ as vertically opposite angles thus making $\angle 2$ = $\angle 4$ . Now, $\angle 4$ and $\angle 6$ are alternate interior angles and thus $\angle 4$ and $\angle 6$ are equal. Thus, we have $\angle 2$ and $\angle 6$ .

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