Euclid,

ElementsI 32©

translated by Henry Mendell (Cal. State U., L.A.)

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(plain diagram)

(general diagram)

Statement

Upon one of the sides of any triangle being extended the external angle is equal to two, the 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.

Determination

I say that the external angle, that by AGD is equal to two, the interior and opposite angles, those by GAB, ABG, and the three interior angles of the triangle, those by ABG, BGA, GAB are equal to two right-angles.

Construction

(diagram 2)

For let a parallel, GE, to straight-line AB be drawn through point G.

Demonstration (part 1)

(diagram 3)

And since AB is parallel to GE and AG falls within them, the alternate angles, those by BAG, AGE, are equal to one another. (diagram
4) Again since AB is parallel to GE, and a straight-line, BD, falls within them, the external angle, that by EGD, is equal to the interior and opposite angle, that by ABG. (diagram
5) But the angle by AGE was shown equal to that by BAG. Conclusion (part 1) Therefore, a whole, the angle by AGD, is equal to two, the interior and opposite angles, those by BAG, ABG.

Demonstration (part 2)

(diagram 6)

Let a common be added, that by AGB. Therefore, the angles by AGD, AGB are equal to three, those by ABG, BGA, GAB. But the angles by AGD, AGB are equal to two right-angles. Therefore, the angles by AGB, GBA, GAB are also equal to two right-angles.

Conclusion (part 2)

Therefore, upon one of the sides of any triangle being extended the external angle is equal to two, the interior and opposite angles, and the three interior angles of the triangle are equal to two right angles, just what it was required to show.