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# The second theorem of the outer angle

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## giulia lorenzi

on 8 May 2015

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#### Transcript of The second theorem of the outer angle

The second theorem of the outer angle
The theorem says...
each outer angle in a triangle is congruent to the sum of the non-adjacent inside angles.

let's demostrate it.
HYPOTHESIS AND THESIS
hp: ABC is a triangle,
the angle ACD is the outer angle of the triangle ABC
the angle CAB= α
the angle ABC= β

th: the angle ACD is congruent to α+β
the second theorem of the outer angle describes exactly the relation between an outer angle and the not adjacent inside angles.

DEMONSTRATION
We've got that:
-the angle ACD is congruent to β'+α' (for construction)
-α is congruent to α' (because alternated inside angles respect to the parallel straight lines of AB and CE, cut by the transversal line AC)
-β is congruent to β' (because correspondent angles respect to the parallels AB and CE, cut by the transversal line BD)

so the angle ACD is congruent to α+β

CONSTRUCTION
let's draw from C the parallel of AB, denoting with α' and β' the angles in which the angle ACD is divided by this straight line.

by Giorgia Fratini and Giulia Bruno ;)
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