Opposite Angles in a Cyclic Quadrilateral Add to 180°
A, B, C and D all sit on the circle, so ABCD is a cyclic quadrilateral. Whatever shape you drag it into, the two pairs of opposite angles each add to 180°:
- ∠DAB + ∠BCD = 180°
- ∠ABC + ∠CDA = 180°
Opposite angles share a colour on the diagram, so the pairing is visible before any arithmetic starts.
How to Use
Interactive Exploration
Drag any vertex around the circumference. Both coloured pairs keep adding to 180° throughout. A vertex cannot be dragged past its neighbours — that would fold the quadrilateral over itself and it would no longer be the shape the theorem is about.
Watch which angles actually move while you drag one vertex: moving A leaves ∠DAB and ∠BCD alone, and changes ∠ABC and ∠CDA together. The pair at the vertex you are dragging stands on the arcs you are not changing, so it holds still — dragging A is the “angles in the same segment” result again, hiding inside this one.
Solving Problems
New positions sets a fresh question. Two angles are hidden, one from each opposite pair — marked x and y. Hiding just one would leave three showing, which is more than the problem needs; this way both pairs get used, and part of the work is deciding which given angle is opposite each unknown.
Tap the x or the y on the diagram, or click the matching reveal, to check an answer. Hovering a reveal lights up the angle it names.
The Style tab changes the pair colours and the background paper.
Teaching Notes
- The arcs are chosen so every angle is a whole number and each pair adds to exactly 180°.
- Why it works: each angle stands on the arc opposite it, so it is half that arc. The two arcs in a pair make up the whole circle, and half of 360° is 180°. It follows straight from the angle at the centre theorem.
- A good prompt once students are confident: “If one angle is 90°, what do you know about the one opposite? What kind of quadrilateral could this be?”
- Careful with the converse — students often assume any quadrilateral has this property. Ask them to try it on a quadrilateral whose corners are not all on the circle.