Angles in the Same Segment are Equal
A and B are the ends of a chord. C and D both sit on the same side of that chord — in the same segment — and both look back at A and B.
The two angles are always equal. Slide C or D anywhere along its arc and the angle it makes does not change at all: the only thing that decides its size is the arc AB it stands on.
How to Use
Interactive Exploration
Drag C or D along the arc. Neither angle changes — this is the part students rarely believe until they see it. Then drag A or B to open or close the chord, and watch both angles change together, staying equal.
C and D are held in the same segment on purpose. If one could cross the chord the angles would no longer be equal — they would add to 180° instead, which is the cyclic quadrilateral result.
Solving Problems
New positions sets a fresh question: one angle keeps its value and the other is replaced by x. The answer is the same number — the point is recognising why, and spotting the two angles that stand on the same arc.
Click ∠ACB or ∠ADB to check. Drag a point while an answer is showing and the value updates with it.
The Style tab changes the arc and chord colours and the background paper.
Teaching Notes
- Everything snaps to 2° steps, so both angles are whole numbers and always exactly equal.
- This follows straight from the angle at the centre theorem: both angles are half of the same angle at the centre, so they must be equal to each other.
- A good prompt: “How many more points could I put on that arc, and what angle would each of them give?”