+0

Help!

+2
31
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+902

A hexagon is inscribed in a circle:

What is the measure of $$\alpha$$, in degrees?

Jan 12, 2019

#1
+313
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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!

Jan 12, 2019
#2
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it only works where the angles are all opposite of each other

Jan 12, 2019
#3
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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.)

Jan 12, 2019
edited by Melody  Jan 12, 2019
edited by Melody  Jan 12, 2019
#4
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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°

Jan 12, 2019
edited by CPhill  Jan 12, 2019
edited by CPhill  Jan 12, 2019
#5
+94995
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You lost me here Chris

"The 110°  angle  intercepts a major arc of 220°"

Melody  Jan 12, 2019