@The-Blade-Dancer You are right, using the Inscribed Angle Theorem would work a lot faster. 🙂 That theorem tells us that we can take the difference of the two arcs that are inscribed by angle \(P,\) and divide that by \(2\) to get the measure of \(\angle P.\)

Screenshot from 2020-11-12 17-15-19.png

$$ \angle P = \frac{80^{\circ} - 30^{\circ}}{2} = \frac{50^{\circ}}{2} = \boxed{25^{\circ}} $$

You've finished the course already and are just reviewing, and so this theorem, which is proved in Day 10, isn't something that the kids know about when they're doing the Day 6 lesson. Maybe that's why Prof. Loh tried to use other methods to solve it! 🙂 That's my guess, but I also know that Prof. Loh likes to use different solution methods to check his work, also. 🙂

@The-Rogue-Blade You can do a shortcut in your head and not worry about the central angle, since the central angle is always the same measure as the arc that it opens up to. So to find the measure of the \( \textcolor{purple}{\text{ angle, }}\) just find the arc that it opens up to, and take half of it. That's all you do: look for the arc.

Thus \( \textcolor{purple}{\text{ angle } \frac{a}{2}}\) is half of the measure of \( \textcolor{purple}{\text{ purple arc a,}}\) and \( \textcolor{orange}{\text{ yellow arc b}}\) is half of the measure of \( \textcolor{orange}{\text{ yellow arc b.}}\)

When I read your question, this made me think back to the proof of the Inscribed Angle Theorem which, at the time, only had the special case of an angle with one side as the diameter of the circle. So I've added to that proof by giving an additional visual proof of the Inscribed Angle Theorem for general angles, including angles with a tangent line, like the right-hand \(\textcolor{orange}{\text{ yellow }}\) angle in the diagram.

M2W2D6-y-part-1-tangent-secant-solution-20-percent.png
 

Please check it out! 🙂

I hope the post with the proof of the Inscribed Angle Theorem will make it more clear why the \(\textcolor{purple}{\text{ purple }}\) angle has measure \( \textcolor{purple}{\frac{a}{2}}\) and both \(\textcolor{orange}{\text{ yellow }}\) angles have measure \(\textcolor{orange}{\frac{b}{2}}\)!

Additionally, this other forum post has a visualization for the formula of an inscribed angle as well as some examples of inscribed angles.