Arc vs Arc Measure

Question

Theorem: The radius of a circle is perpendicular to the tangent at the point of tangency

One particular proof I came across is as follows:

$$\text{The measure of} \: \: \angle CAB \: \: \text{is one half the arc it cuts off. Since the chord is the diameter, the arc is half the circle, so } \angle CAB = \frac{180^{\circ}}{2}$$

However here $180^{\circ}$ is not the length of the arc. It is the measure of the angle that subtends it.

Similarly stated:

The measure of an inscribed angle is half the arc it subtends

It is stated that $\angle ABC = \frac{\frown {AC} }{2}$

But shouldn't it be $2r\angle ABC = \frown AC \implies \angle ABC = \frac{\frown AC}{2r}$

Answer 1

Those quotes are misleading, they make one think that an arc is measured by arc length. But that's not what's going on in this context. Adding a few words to those quotes might clear things up:

• The angle measure of an inscribed angle is half the angle measure of the arc it subtends.

You must take care, of course, that these angles measures use the same units: you could use units of degrees; or instead use units of radians.

So, for instance, the angle measure in degrees of a semicircular arc is indeed $180^\circ$, which equals half the angle measure in degrees of the whole circle which is $360^\circ$. And so $\angle CAB$ does indeed have an angle measure in degrees equal to $\frac{180^\circ}{2} = 90^\circ$.

Answer 2

An arc is a piece of a curve in the euclidean plane. The measure of this arc is its length. If the arc is a piece of a circle then the length of the arc, divided by the radius of the circle, is the natural measure for the angle subtended at the center of the circle.

In your figures it is tacitly assumed that the drawn circle has radius $1$. – By the way: There must be proofs of said theorem that rely on more basic facts about circles and tangency.