MHB Prove 1 & 2: Bisector Properties of Angles

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An angle has only one bisector because it corresponds to a unique value of half the angle, meaning there cannot be multiple lines that equally divide the angle. The discussion highlights the importance of clarity in defining angles and their bisectors, suggesting that diagrams could aid understanding. It emphasizes that if an angle were to have more than one bisector, it would contradict the fundamental property of angle division. The analogy of halving a number illustrates that just as a specific number has one unique half, an angle similarly has only one bisector. Therefore, the properties of angles ensure that each angle is bisected uniquely.
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How can (1) I prove that there is only one bisector of a angle to only one specific angle
and
(2)There is only a specific angle with only one bisector of it.
 
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I think your question needs clarification - perhaps a diagram? Can you be more specific? As it is, a line may bisect infinitely many angles, I believe.
 
What will happen if a angle has more than one bisector?
What will happen if some angles have more than one bisectors and other not?

What is the reason that angle has only one bisector?
 
highmath said:
What is the reason that angle has only one bisector?

An angle $\alpha$ has only one bisector because there is only one possible value for $\frac{\alpha}{2}$.
 
It seems to me to be similar to asking if, for any particular number, are there multiple values that are half of that number. For example, we know 6 is half of 12...can you think of another number that is half of 12?
 

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Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
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