Question 293120: I have been working on this math problem and I can't seem to figure it out. I was wondering if someone could help me? Please and Thank You! I would deeply appreciate it!
Circles: other angles
A quadrilateral circumscribed about a circle has angles of 80 degrees, 90 degrees, 94 degrees, and 96 degrees. Find the measures of the four nonoverlapping arcs determined by the points of tangency.
Answer by Earlsdon(6294)   (Show Source):
You can put this solution on YOUR website! The key here is noting that the angle formed by the circle's radii to the points of tangency is the supplement of the angle formed by the two rays from the corresponding corner of the quadrilateral.
So if you were to accurately draw the quadrilateral and the circle, then draw the radii to the four pounts of tangency formed by the four sides of the quadrilateral with the given angles (80, 90, 94, and 96 degrees), you would find that the angle formed by the pairs of radii would be equal to the supplement of the angle of the corresponding corner of the quadrilateral.
I hope I made that clear.
Anyway, you can now determine the measure of the four arcs formed by the points of tangency.
The four angles inside the circle are:
1) 180-80 = 100 degrees.
2) 180-90 = 90 degrees.
3) 180-94 = 86 degrees.
4) 180-96 = 84 degrees.
...and not surprisingly, these sum to 360 degree.
To find the measure of each arc, divide the interior angle by 360 and multiply by the circle's circumfernce ( ).
1) 
You can work out the others using this method.
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