Interior and Exterior AnglesDate: 11/01/2001 at 13:33:49 From: Anonymous Subject: Geometry I have calculated that the value of an angle in a quadrilateral is 65 degrees. The other angles are 12, 104, and 27 degrees, and the sum is equal to 208 and not 360. Is this correct? Date: 11/02/2001 at 12:02:33 From: Doctor Peterson Subject: Re: Geometry Hi, I suspect that you have a non-convex quadrilateral, and one of your angles is not an INTERIOR angle. It's the interior angles that add up to 360 degrees. In particular, if the shape looks something like + / / / 65/ / / /12 / +------+ / 104\27/ \ / + then the interior angle corresponding to 104 degrees is 360 - 104 = 256, and the sum of interior angles is 12+256+27+65 = 360 as expected. Now, how did I identify the 104 degree angle as the culprit? Suppose in general that you have a quadrilateral with normal angles A, B, and C, but that angle D was measured on the outside, so that the interior angle is really 360-D. Then the sum of interior angles is A + B + C + (360-D) = 360 and A + B + C = D To find which angle was measured on the outside, you can compare the sum you got with what you expected: sum = A+B+C+D = (A+B+C) + D = 2D so the mismeasured angle is just half of the sum you got. Half of 208 is 104, which is why I chose that as the reflex angle in my example. It's fun being a Math Detective! Now make sure you always measure interior angles in the future, so I won't have to arrest you! - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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