# Elements: Book I, Proposition 6

Math Lair Home > Source Material > Elements > Book I, Proposition 6

The following is as given in Sir Thomas L. Heath's translation, which can be found in the book The Thirteen Books of The Elements, Vol. 1 .

Elements on the Math Lair
Book IBook IIBook IX
Definitions, Postulates, and Common NotionsDefinitionsProposition 20
Proposition 1, Proposition 3, Proposition 14Proposition 36
Proposition 5, Proposition 6,
Proposition 29, Proposition 47

## Proposition 6.

If in a triangle two angles be equal to one another, the sides which subtend the equal angles will also be equal to one another.

Let ABC be a triangle having the angle ABC equal to the angle ACB; I say that the side AB is also equal to the side AC.

For, if AB is unequal to AC, one of them is greater.

Let AB be greater; and from AB the greater let DB be cut off equal to AC the less;

let DC be joined.

Then, since DB is equal to AC,

and BC is common,

the two sides DB, BC are equal to the two sides AC, CB respectively;

and the angle DBC is equal to the angle ACB;
therefore the base DC is equal to the base AB,

and the triangle DBC will be equal to the triangle ACB,

the less to the greater;

which is absurd.

Therefore AB is not unequal to AC;

it is therefore equal to it.

Therefore etc.

Q.E.D.