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; [I. 3] let DC be joined. [Post. 1]

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;

[I. 4]therefore the base DC is equal to the base AB,

and the triangle DBC will be equal to the triangle ACB, [I. 4]

[C.N. 5]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.