In the context of angles in standard position, which of the following pairs of angles are adjacent to each other?

• A. $30°$ and $60°$
• B. $90°$ and $180°$
• C. $45°$ and $135°$
• D. $30°$ and $120°$ that share the same vertex and one common side but do not overlap

Answer: (D)

Verified step by step guidance

1. Recall the definition of adjacent angles in standard position: two angles are adjacent if they share the same vertex, have one common side, and their interiors do not overlap.
2. Identify the vertex for all given angles, which is the origin in the coordinate plane for angles in standard position.
3. Check each pair to see if they share a common side (ray) starting from the vertex. For example, angles 30° and 60° have rays at 30° and 60°, so they do not share a side.
4. Verify if the difference between the two angles is exactly the measure of the angle between their rays. Adjacent angles will have rays that coincide on one side, meaning the difference between their measures corresponds to the other side.
5. Conclude that the pair 30° and 120° are adjacent because they share a common side (the ray at 30°) and their interiors do not overlap, fulfilling the criteria for adjacency.