Book 1
5
Accordingly, since the whole angle ABG was proved equal to the angle ACF, and in these the angle CBG is equal to the angle BCF, the remaining angle ABC is equal to the remaining angle ACB; and they are at the base of the triangle ABC. But the angle FBC was also proved equal to the angle GCB; and they are under the base.
5
Therefore etc.
5
Q. E. D.
Proposition 6.
6
Enunciation 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.
6
Proof. Let ABC be a triangle having the angle ABC equal to the angle ACB;
6
I say that the side AB is also equal to the side AC.
6
For, if AB is unequal to AC, one of them is greater.
6
Let AB be greater; and from AB the greater let DB be cut off equal to AC the less;
6
let DC be joined.
6
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.
6
Therefore etc.
6
Q. E. D.
Proposition 7.
7
Enunciation Given two straight lines constructed on a straight line (from its extremities) and meeting in a point, there cannot be constructed on the same straight line (from its extremities), and on the same side of it, two other straight lines meeting in another point and equal to the former two respectively, namely each to that which has the same extremity with it.
