Fundamental Concepts
18. The Adjacent Angle Rule · a common arm, a common vertex, and no overlap
The rays from O always form a pair of adjacent angles — they stop being adjacent only when two rays overlap.
Adjacent angles are angles that have a common arm and a common vertex, but do not overlap. The shared arm lies between them, so the two angles sit side by side and neither one falls inside the other.
What this lesson covers
Try to break it
Try to break the pairing: drag B and C until rays OB and OC land on top of each other. Once two rays overlap, the figure no longer shows a clean pair of adjacent angles.
How you build it
Make two adjacent angles that share the arm OA.
- Click the Point tool and place vertex O.
- Place point A — this will be on the shared ray OA.
- Click the Ray tool, click O then A to draw ray OA.
- Place point B on one side of OA.
- Draw ray OB (Ray tool: O then B). Now ∠AOB exists.
- Place point C on the OTHER side of OA — B and C must lie on opposite sides.
- Draw ray OC (Ray tool: O then C). ∠AOB and ∠AOC share ray OA and sit on opposite sides — they're adjacent.
The proof, step by step
Prove that ∠AOB and ∠AOC are adjacent angles.
- ∠AOB and ∠AOC share the same vertex, O.
- ∠AOB and ∠AOC share the common arm OA.
- Arms OB and OC lie on opposite sides of OA, so the two angles do not overlap.
- Therefore, ∠AOB and ∠AOC are adjacent angles.
Worked example
Which statement correctly defines adjacent angles?
Adjacent angles have a common arm and a common vertex, and they do not overlap — neither angle falls inside the other.
- Two angles that have a common arm and a common vertex, and do not overlap. — correct
- Two angles that share a vertex but have no common arm.
- Two angles that share an arm but have different vertices.
- Two angles whose interiors completely overlap each other.