The problem provides a diagram with two parallel lines $g$ and $h$, crossed by two transversals $i$ and $k$, forming angles 1, 2, 3, and 4. You are asked to identify which statement about these angles must always be true.

The key concept here involves properties of angles formed when parallel lines are intersected by a transversal:

1. Alternate Interior Angles: Angles on opposite sides of the transversal but inside the parallel lines are equal.
2. Corresponding Angles: Angles in the same position relative to the parallel lines and transversal are equal.
3. Consecutive Interior Angles: Angles on the same side of the transversal and inside the parallel lines are supplementary (sum to 180°).

Evaluating the Options:

• Option A: "If $\angle 1 \cong \angle 2$, then $g \parallel h$."

  • False: $\angle 1$ and $\angle 2$ are on the same side of the transversal but not corresponding angles. They are consecutive interior angles, so they should be supplementary, not congruent, for $g \parallel h$.

• Option B: "If $\angle 2 \cong \angle 3$, then $g \parallel h$."

  • False: $\angle 2$ and $\angle 3$ are alternate interior angles. They must be equal if $g \parallel h$, but the reverse isn't necessarily true; other configurations could also make them equal.

• Option C: "If $\angle 2 \cong \angle 4$, then $j \parallel k$."

  • False: $\angle 2$ and $\angle 4$ are on opposite sides of the transversal $i$, and knowing they are congruent wouldn't establish $j \parallel k$.

• Option D: "If $\angle 3 \cong \angle 4$, then $j \parallel k$."

  • True: $\angle 3$ and $\angle 4$ are corresponding angles. If these angles are congruent, it means that lines $j$ and $k$ must be parallel.

Correct Answer:

The correct statement that must always be true is Option D: "If $\angle 3 \cong \angle 4$, then $j \parallel k$."

Would you like a detailed explanation on why the other options are incorrect? Here are five related questions to consider:

1. What are alternate interior angles, and why are they important when proving lines parallel?
2. How can you prove two lines are parallel using corresponding angles?
3. What is the significance of supplementary consecutive interior angles?
4. Can two lines be parallel if only one pair of corresponding angles is equal?
5. How does the relationship between parallel lines and a transversal affect the angles formed?

Tip: Always check if angles that are supposed to be equal or supplementary are in the correct corresponding positions when dealing with parallel lines.