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A hexagon is inscribed in a circle:

What is the measure of \(\alpha\), in degrees?

Lightning Jan 12, 2019

#1**+2 **

the figure is cyclic, so those three angles are just half of the total sum of all the angles, which is 720/2. subtracting, you get 360-105-110=145.

HOPE THIS HELPED!

asdf335 Jan 12, 2019

#3**+2 **

Join the alpha vertex to the vertex opposite and then you have two cyclic quads. (one on top and one underneath)

So alpha is the sum of (180-105)+(180-110) = 75+70=145 degrees

I know this is the same as asad's answer, it is just another way of doing it (I actually did not understand asad's method.)

Melody Jan 12, 2019

#4**+2 **

Thanks asdf and Melody......here's another way to look at this....but not as nice as yours....!!!

The 110° angle intercepts a major arc of 220°......so the minor arc that it subtends = 360 - 220 = 140°

The 105° angle subtends a major arc of 210° ......so the minor arc it subtends = 360 - 210 = 150°

So.....this means that the minor arc subtended by "a" must be 360 - 140 - 150 = 70°

So.....the major arc it intercepts = 360 - 70 = 290°

So....its measure is 1/2 of this = 145°

CPhill Jan 12, 2019