Proving the inscribed angle theorem

Question

I need to prove that a circle's inscribed angle is 1/2 of the arc it intercepts. I am given that one of the chords making up the angle is the diameter. I have an entire project to do based off of this proof, so I really need to prove this.

Answer 1

The case you drew is perhaps the easiest one:

$\;\Delta AOC\;$ is isosceles, with $\;AO=OC\implies \angle ACO=\angle CAO\;$, and since

$\;\angle AOB\;$ is an external angle to triangle $\;\Delta AOC\;$ , then it equals the sum of the two triangle's

angles whose vertex it doesn't share, thus

$$\angle AOB=\angle CAO+\angle ACO=2\angle ACO$$

Answer 2

Hint:
If $\angle AOB=2\alpha$, then what is $\angle AOC$? $\angle ACO$?