In this interactivity, you can move the points \(A\), \(B\), \(C\) and \(D\) around the circumference of the circle, with \(D\) between \(B\) and \(C\).

A circle with points A, B, D, C in this order on it.
  1. Three angles are marked.

    What can you say about the angles at \(C\) and \(D\)?

    What happens to these angles if you move \(C\) or \(D\), keeping \(A\) and \(B\) fixed?

    How do you know this?

We have labelled the angle at \(B\) as \(\theta\). This makes it easier to talk about it.

What theorems do you know about angles in circles? Are any of them relevant here?

  1. What happens to the exterior angle at \(B\) (shown in blue) as \(D\) approaches \(B\)?

Can you work out the size of this angle, \(\theta\), in terms of other angles?

  1. Which other circle theorem can you deduce from this?

What else happens as \(D\) approaches \(B\) – what does the limiting case look like?