Equivalence of Definitions of Tangent of Angle

Theorem
Let $\theta$ be an angle.

Then the following definitions of tangent of $\theta$ are equivalent:

Definition from Triangle implies Definition from Circle
Let $\tan \theta$ be defined as $\dfrac {\text{Opposite}} {\text{Adjacent}}$ in a right triangle.

Consider the triangle $\triangle OAB$.

By construction, $\angle OAB$ is a right angle.

Thus:

That is:
 * $\tan \theta = AB$

Definition from Circle implies Definition from Triangle
Let $\tan \theta$ be defined as the length of $AB$ in the triangle $\triangle OAB$.

Compare $\triangle OAB$ with $\triangle ABC$ in the diagram above.

We have that:
 * $\angle CAB = \angle BOA = \theta$
 * $\angle ABC = \angle OAB$ which is a right angle

Therefore by Triangles with Two Equal Angles are Similar it follows that $\triangle OAB$ and $\triangle ABC$ are similar.

By definition of similarity:

That is:
 * $\dfrac {\text{Opposite} } {\text{Adjacent} } = \tan \theta$