(general diagram)
Statement
Of every triangle, when one of the sides is extended the external angle
is equal to the two interior and opposite angles and the three interior
angles of the triangle are equal to two right angles.
Display
(diagram 1)
Let there be a triangle ABG and let one side of it BG be extended to
D. I say that the external angle under AGD is equal to the two interior
and opposite angles under GAB, ABG and the three interior angles under
ABG, BGA, GAB are equal to two right angles.
Construction
(diagram 2)
For let GE be drawn through G parallel to straight-line AB.
Demonstration (part 1)
(diagram 3)
And since AB is parallel to GE and AG falls on them, the alternate
angles under BAG, AGE are equal to one another. (diagram
4) Again since AB is parallel to GE, and straight-line BD falls on
them, the external angle under EGD is equal to the interior and opposite
angle under ABG. (diagram
5) But the angle under AGE was proved equal to the angle under BAG.
Therefore, the whole, the angle under AGD, is equal to the two interior
and opposite angles under BAG, ABG.
Demonstration (part 2)
(diagram 6)
Let the common angle under AGB be added. Therefore, the angles
under AGD, AGB are equal to the three under ABG, BGA, GAB. But the
angles under AGD, AGB are equal to two right angles. Therefore the
angles under AGB, GBA, GAB are equal to two right angles as well.
Therefore, of every triangle, when one of the sides is extended the external angle is equal to the two interior and opposite angles and the three interior angles of the triangle are equal to two right angles, just what it was required to prove.