Mike K.
asked • 04/22/17opposite angles of an inscribed hexagon in a circle
I know that opposite angles of a quadrilateral inscribed in a circle are supplementary. What about opposite angles of an inscribed hexagon in a circle? It should be 240, but can you show help me with the proof?
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2 Answers By Expert Tutors
Kenneth S. answered • 04/22/17
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Let's say that you have a hexagon inscribed in a circle, i.e. the six vertices are all points on the circle.
For identification purposes, label consecutive angles 1, 2, through 6. Consider angle 1. Adjacent angles are 2 and 6; what would you consider to be the angle opposite Angle 1? (3, 4 or 5?)
So you see that the concept of opposite angles, meaningful for an inscdribed quadrilateral, breaks down (is inapplicable) for a ngon where n>4.
Mike K.
Put the comment in the wrong section.
Opposite would be 'directly' opposite, hence 4. I'm looking for the sum of angles 1 & 4 (2 & 5, or 3 & 6) in your example.
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04/23/17
Mark M. answered • 04/22/17
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Internal angle of a regular of a regular polygon is 180(n - 2) / n, where n is the number of sides
For a hexagon, 180(4)/6, or 120°
Mike K.
I should've been clear - he 'sum' of directly opposite angles is 240.
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04/23/17
Mark M.
The hexagon has three pairs of opposite angles. Each angle in the pair is 120°. 120° + 120° = 240°
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04/23/17
Mike K.
That is what I stated. I'm looking for the 'proof' that sum of directly opposite angles in a hexagon is 240. This would be for a non-regular hexagon inscribed in a circle. Much like a similarly inscribed quadrilateral where the sum of opposite angles is 180 when inscribed in a circle.
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04/23/17
Kenneth S.
Everyone, think about an inscribed pentagon. How does the concept of opposite angles work for you in this case?
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04/23/17
Mark M.
No such proof exists. For confirmation you can query on math.stackexchange,com.
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04/23/17
Kenneth S.
No such proof exists because of the ambiguity, in general, of opposite angles for inscribed n-gons.
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04/23/17
Mike K.
Actually, I just realized that it shouldn't the sum of angles A and D (or B & E, or C & F), but it should be the sum of angles A,C,E (or B,D,F) that's constant. That sum adds to 360. Once I realized that, it was fairly easy. My bad for not stating the problem incorrectly. It's true for any hexagon inscribed in a circle.
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04/23/17
Mark M.
Even with your correction. It is impossible to prove that all three pairs of opposite angles add to 360°. Using a graphing application I have constructed a irregular hexagon in which all pairs of opposite angels do not add the 360°.
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04/23/17
Mike K.
muse a circle theorem - where opposite angles in inscribed quadrilateral sum up to 180. Apply that to any hexagon. You sum angles A, C, & E and get 360. Use proof not scaled drawings.
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04/23/17
Mark M.
How true, opposing angles of a quadrilateral sum to 180°. Yet you only have two pair of such angles. You keep changing the ground rules. One cannot proof that something cannot be proved. That is logically ridiculous.
I note that you have posited this same series of questions on math.stackexchange.com. Perhaps you could devote your time, and the time of the tutors on this site, to more product discussion/inquiry.
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04/23/17
Mike K.
The problem wasn't stated to me clearly, so the part of the solution was to understand what the problem was getting at. As I found out, I provided more details and clarification. I clearly stated that A,C,E add to 360, which you didn't seem to understand based on your response. It's fine if you didn't understand the problem clearly. I don't understand why you're getting so defensive. I posted in other forums because I wanted to cast a wider net, and from one of them I heard back that gave me the hint on what the problem was about to begin with. Also, you're the one that suggested on of them, are you not? Yes, one of them carried on discussion with me, via email, and we had wonderful discovery. Why do you seem so upset?
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04/23/17
Mark M.
You seem to conflate my justifiable criticism with some degree of emotional involvement.
Even the addition of any three non-sequential angles may or may not add to 360°. I have constructed several to demonstrate that. This site does not allow me to share them with you. Perhaps you might do come empirical investigation.
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04/23/17
Mike K.
I'll state it once again - given a hexagon inscribed in a circle, ABCDEF, the angle sum of A+C+E (or, B+D+F) will give you 360 degrees. A+C+E (or B+D+F) is critical in that it is a sum of every second angle, not just any three angles, hence how it was stated previously. It can be proven using more than one method. Method 1) by looking at the sum of three inscribed angles, A,C,E in terms of central angles. You get 2*(A+C+E) = 720. => A+C+E = 360. Method 2) splitting the hexagon into two quadrilaterals and using inscribed quadrilateral theorem. That needs to be done three times. Draw a line from point A to point D, where angle A consists of a1 and a2. a1 + E = 180 & a2 + C = 180. Draw a line from point C to F, where angle C consists of c1 and c2. c1 + E = 180 & c2 + A = 180. Draw a line from point E to B, where angle E consists of e1 and e2. e1+ C = 180 & e2 + A = 180. Add these six equations, LHS = 3(A+C+E). RHS = 180*6. => A+C+E = 360.
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04/23/17
Mark M.
A, C, and E would not be central angles. They would be inscribed angles equal to one-half of the subtended arc.
I shall study the second method tomorrow.
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04/23/17
Mike K.
A,C,E are inscribed angles. You can express the sum of these inscribed angles m in terms of central angles in the proof.
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04/23/17
Mark M.
I stand corrected and informed and applaud your reasoning
1)
A = 0.5 arc BF
E = 0.5 arc FD
C = 0.5 arc DB
A + C + E = 0.5(720)
A+ C + E = 360
2)
I drew a sketch of your directions/proof. Enlightening!
Thank you for your patience and willing to share this with me.
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04/24/17
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Mike K.
04/23/17