Exercise 3 Page 213

Let's consider each of the given options one at a time.

$\angle 1\cong \angle 4$

From the diagram, we can tell that $\angle 1$ and $\angle 4$ are corresponding angles.

Let's recall Converse of the Corresponding Angles Theorem. Since we are given that angles $\angle 1$ and $\angle 4$ are congruent, we can use this theorem to prove that lines $\ell$ and $m$ are parallel.

$\angle 2\cong \angle 5$

From the diagram, we can tell that $\angle 2$ and $\angle 5$ are alternate interior angles.

Let's recall Converse of the Alternate Interior Angles Theorem. Because we are told that angles $\angle 2$ and $\angle 5$ are congruent, we can use this theorem to conclude that lines $\ell$ and $m$ are parallel.

$\angle 3\cong \angle 4$

In the diagram, we see that $\angle 3$ and $\angle 4$ are also alternate interior angles. Once again, we are told that they are congruent.

Therefore, similar to option (II), we can use the Converse of the Alternate Interior Angles Theorem to prove that the lines are parallel.

$m\angle 2+m\angle 4=180$

One last time, looking at the diagram, we can tell that $\angle 2$ and $\angle 5$ are same-side interior angles.

Let's recall Converse of the Same-Side Interior Angles Postulate. Since the sum of angles' measures is $180^{\circ },$ they are supplementary. Therefore, using this postulate, we can prove that lines $\ell$ and $m$ are parallel.

Conclusion

As we can see, each option gives us a piece of information that allows us to prove that the lines $\ell$ and $m$ are parallel. Therefore, the answer is D.