A, B and C are points on the circle, centre O. DE is a tangent to the circle at A. Angle AOC = $4x^\circ$ and angle ACB = $x^\circ$. Find, in terms of x, the following. Give a reason for each step of your answer.

1. Relate the centre angle to the circumference angle standing on the same arc $AC$.

 AOC = 2 ABC

Reason: angle at the centre is twice the angle at the circumference standing on the same arc $AC$.

2. Substitute $ AOC=4x^3$ and find $ ABC$.

4x = 2 ABC

 ABC = 2x^3

1. Identify the angle in the alternate segment corresponding to tangent $AE$ and chord $AB$.

$AE$ is a tangent at $A$ and $AB$ is a chord.

The angle between tangent $AE$ and chord $AB$ is equal to the angle in the alternate segment, which is $\angle ACB$.

2. Substitute the given value of $\angle ACB$.

Since $\angle ACB = x^\circ$,

$\angle BAE = \angle ACB = x^\circ.$