@The-Rogue-Blade

Ah! Good question! It's because you want to look for the "X" in the diagram. The "X" will tell you which two angles go together in the equation given in the video.

M2W2D6-y-part-2-triple-chords-solution-why-not-80-a-and-b.png
 

The arcs \((80 - a)\) and \(a\) are both inscribed by the "X." This is why we found that the angle \(40^{\circ}\) was the average of \(a\) and \(80-a.\)

 
M2W2D6-y-part-2-triple-chords-solution-10-percent.png

 
 
However, the arc \(b\) and the arc \((80-a)\) aren't inscribed by two chords which intersect in an "X."
 

M2W2D6-y-part-2-triple-chords-solution-not-80-a-and-b.png

Instead, the arc \(b\) pairs with the arc \(c,\) to form an "X." Thus, the following would be true:

$$ \frac{b + c}{2} = 70^{\circ} $$

🙂

@TSS-Graviser There isn't much of a big "reason", it's simply how we define the "measure" of an "arc".

If you think about it, arcs aren't really "angles", right? But, we find that it is very useful to just "say" that an arc has an "angle measure". But why, and what would it even represent?

An arc is just part of a circle. So, we can think of the "angle measure" of an arc as how much the arc goes around the circle. For example, the full circle is 360 degrees - if you stood on a playground and drew a full circle around you with a stick, you'd spin a full "360 degrees". If you only went a quarter of the way around, you only draw "90 degrees" of an arc.

As you can imagine, how much you go "around" is exactly the central angle you make!

72888ed9-7bcc-40ef-aef1-531f11fd12b0-image.png

So why is this useful? Off the top of my head, a good reason is that using "arcs" instead of "central angles" can be a good shortcut, and can be easier to think about. For example, inscribed angles along the same arc have the same measure!

db49cd76-307c-4cdd-b311-750e921f0e38-image.png

And we can say that the measure of each of those inscribed angles is half the measure of the intercepted arc, and we don't have to draw in the central angle.

Yes, that's right! It's very good to be thinking, "But does this only work for this diagram? What if we were to draw the intersecting chords differently?" That's why Prof. Loh tries to prove or illustrate why every theorem is true, so you can see when it can be applied. The proof in this case can be seen in the diagram from the video:

39fb8db3-7e12-4087-999a-b8f9fc4b3d1e-image.png

You can see that the angle that is formed from the intersection of the chords is equal to

$$ \frac{a + b}{2} $$

This is true because this angle is the exterior angle of the triangle containing the green line. Use the fact that the exterior angle of a triangle is equal to the sum of the other two angles of the triangle.

The great thing about trying to prove why identities and theorem are true is that it helps you remember them, also. And it helps you to analyze new situations with a clever, creative approach. Great job on thinking about this idea!

Happy Learning,

The Daily Challenge Team