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Q: What are the common endpoint of 2 rays that form an angle?

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An angle.

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An angle is the union of two rays with a common endpoint.

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Two rays that share an endpoint form an angle.

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an angle precisely defined as the common endpoint of two rays

-- Two rays with a common endpoint form an angle. -- If they have a common endpoint, then they are collinear. So I guess, technically, the conditions described in the question aren't possible.

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