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.

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 \(80-a.\)

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

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

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

]]>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!

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!

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.

]]>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

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