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Derivative of implicite function.

Let consider the equation $ F(x,y)=x^2+y^2-1=0 $
We know that the set of solution of this equation is the set of points in plane at distance 1 from the origin, or a circle of radius 1 with center in (0,0).
We want to give a description of this set depending on a single variable instead of two.
This is possible only local ,not for the whole set of solutions.
But the collection of local solutions can give us a complete information about the circle.
Let $ (a,b)\in  R^2 $ with $ a^2+b^2-1=0 $ and $ b\neq 0 $
For $ (x,y) $ in a small neighborhood of $ (a,b) $ we have $ y\neq0 $

In a triangle the sum of the angles is 180

In any triangle the sum the angles is 180.

In triangle ABC:
$ \angle ABC + \angle BCA + \angle CAB = 180.  $

Proof
Let consider line DE by A parallel to BC
Then $ \angle DAC=\angle ACB $ and $ \angle EAB=\angle ABC $ but $ \angle  DAC+\angle CAB +\angle EAB=180 $

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Definition of Angle

An angle is formed by two rays that have a common origin.

Two intersecting lines form 4 angles. The opposing ones are congruent and the neighboring ones have the sum 180 (are supplementary angles).

AEC = DEB
AED = CEB

AEC + AED = 180
CEB + BED = 180
AEC + CEB = 180
DEA + BED = 180

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