Cyclic Quadrilaterals: Understanding Angle Equality and Ptolemy's Theorem

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Cyclic quadrilaterals exhibit angle equality due to the property that angles subtended by the same arc are equal. The discussion highlights confusion regarding the equality of specific angles depicted in images related to Ptolemy's Theorem. It clarifies that angles in the same segment of a circle are equal, supported by the central angle theorem, which states that the angle at the center is twice that at the circumference. Constructing lines from the segment's endpoints to the circle's center can aid in visualizing this relationship. Understanding these principles is essential for grasping the properties of cyclic quadrilaterals.
Buri
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Well this isn't a homework question (I'm just trying to refresh my memory from the plane geometry I did in high school) and so, I was reading through cyclic quadrilaterals on wikipedia and I don't see how certain angles are equal. Here are two images taken from wikipedia:

http://upload.wikimedia.org/wikipedia/commons/d/d1/Ptolemy's_theorem.svg

http://upload.wikimedia.org/wikipedia/en/8/8b/Ptolemy_sine_proof.svg

How are the blue angles equal? Or the theta 2 equal?

I've tried writing things in different way and extending lines, but to no avail. Any help?
 
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They are angles in the same segment of the circle. And angles in the same segment are equal.

You can prove this by using constructing lines from the end of the segment-chord to center of the circle. Then use the center theorem for both.
 
I agree with the legend, you can see that they're equal if you're happy with the "angle at center is twice angle at circumference."
 

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