Angles of intersecting tangent lines of a circle

Question

It's best that I use a diagram to illustrate the question:

Is there any interesting relationship, one that can be expressed mathematically, between $\angle CDB$ and $\angle CAB$

Answer 1

Unfortunately, this is rather elementary. And it's just actually: $$\angle CDB=180^\circ-\angle CAB$$

Answer 2

As your answer above states, $\angle CDB=180^\circ-\angle CAB$. To derive this, we can construct kite $ABDC$. $m\angle DCA$ and $m\angle DBA$ are both equal to $90^\circ$, being radii to points of tangency. Since circumscribed angle $\angle CDB$ is opposite $\angle CAB$ in kite $ABDC$ which are between pairs of congruent sides (i.e., AC = BA and CD = BD), $\angle CAB$ is supplementary to $\angle CDB$.

One interesting result of this is that the measure of the major arc $BC$ is equal to $180^\circ$ + $m\angle CDB$.

This is because the measure of minor arc $BC$ is equal to $m\angle CDB$, $360^\circ$ - $m\angle CDB$ gives the measure of major arc $BC$. These relationships hold true for any circumscribed angle of any circle.