Euclid, Elements I 6©
translated by Henry Mendell (Cal. State U., L.A.)

Return to Vignettes of Ancient Mathematics
Return to Elements I, introduction
Go to prop. 5 Go to prop. 7

Prop. 6: If two angles of a triangle are equal to one another, then the sides subtending the equal angles will be equal to one another.

(general diagram)

Diagrams for book 1 proposition 6, following the display, construction, and demonstration(diagram 1) Let there be a triangle, ABG, having the angle by ABG equal to the angle by AGB. I say that a side, AB, is also equal to a side, AG. (diagram 2) For if AB is unequal to AG, one of them is larger. Let AB be larger and let from the larger, AB, an equal, DB, to the smaller, AG, be taken away, (i 3) and let DG be joined.
(diagram 3) And so since DB is equal to AG and BG is common, in fact, two, DB, BG, are respectively equal to two, AG, GB, and an angle, that by DBG, is equal to an angle, that by AGB. Therefore, a base, DG, is equal to a base, AB, and triangle DBG will be equal to triangle AGB, the smaller to the larger, which is absurd. Therefore, AB is not unequal to AG. Therefore, it is equal. Therefore, if two angles of a triangle are equal to one another, then the sides subtending the equal angles will be equal to one another, just what it was required to show.

top