a theorem

That day a friend asked me about alternate angle theorem, im going to post it here.

 

The theorem is called

Alternate Segment Theorem

The alternate segment theorem states that an angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.

In simple words: a = b (refer to the diagram above).

So my friend asked me: how to prove that a=b
First, we draw an isosceles triangles inside the bigger triangle with a point touching the center. Then we split the isosceles triangle into two parts, each with angle “y”.
As we all know, the line from the tangent to the center of the circle is 90º, then a+x = 90º.
On the other hand, y+x = 90º
 
From that, we can deduce y+x = a+x
Hence, cancle off the x, we get a=y.
 
b=y, this is because the angle at the center is two times the angle at the circumference, with the same arc. since the angle at the center is 2y, then b = 1/2(2y) = y.
 
a=y, b=y. Conclusion a=b.

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