Warm-up Solution

two angles from a chord in a circle

Try to label as many angles in the diagram above as you can, using the fewest possible number of unknowns to represent them.

We have two angles at \(B\) and \(C\) that are subtended by the same arc, so the angles at \(B\) and \(C\) must be equal.

Also we have triangle \(ABD\) which has its longest side as the diameter of the circle, meaning angle \(BAD\) is a right-angle.

angles B and C are equal, angle A is a right-angle

Just as the angles at \(B\) and \(C\) are the same, then so are the angles \(BAC\) and \(BDC\). If we label our two different angles \(\theta\) and \(\alpha\) then we have the following.

labelling angles

From here we can now write all the angles in terms of \(\theta\) and \(\alpha\) using our knowledge of triangles, to give us;

final solution