The problem shown involves determining whether there is enough information to conclude that line $h$ is perpendicular to line $j$, and which theorem among Theorem 3-1, 3-2, 3-3, or 3-4 supports this conclusion.

Analyzing the Diagram:

• The angle between lines $h$ and $j$ is labeled as $90^\circ$, which typically indicates perpendicularity.
• The labels $12$ and $13$ next to the angle imply two segments adjacent to the right angle, likely suggesting some equality or relationship that is used in a specific theorem.

Possible Theorems:

While I don’t have the exact statements of Theorem 3-1 through 3-4, common geometric theorems that would apply include:

• Perpendicularity Theorem: If two lines form a right angle (as labeled by a $90^\circ$ symbol), they are perpendicular.
• Theorem on Congruent Segments: Could involve confirming perpendicularity using properties such as congruent adjacent segments or sides.

Based on typical naming:

• Theorem 3-1 might relate to a general perpendicularity test.
• Theorem 3-2 could relate to properties of angles formed between intersecting lines.
• Theorem 3-3 might involve congruent segments or triangles.
• Theorem 3-4 often involves properties about the converse of perpendicular or angle theorems.

Given the angle is $90^\circ$, Theorem 3-2 (Yes, Theorem 3-2) seems likely if it directly involves perpendicular lines. This could confirm that $h \perp j$ by the presence of the right angle.

Would you like a deeper explanation or have any specific questions about perpendicularity or related theorems?

Questions for Further Exploration:

1. What are the exact statements of Theorems 3-1 to 3-4, and how do they differ?
2. Can perpendicularity be determined without directly using a $90^\circ$ angle?
3. How would congruent segments impact the decision if the angle were not labeled?
4. What other scenarios would Theorem 3-3 or 3-4 apply to this configuration?
5. Why is a $90^\circ$ angle essential in proving perpendicularity?

Tip:

When working with diagrams, ensure you verify angle measures directly on the diagram to support your conclusions.