*EUCLID'S ELEMENTS*.

For if the triangle *ABC* be applied to the triangle *DEF*, so that the point *A* may be on the point *D*, and the straight line *AB* on the straight line *DE*, the point *B* will coincide with the point *E*, because *AB* is equal to *DE*,

[*Hyp*.

And, *AB* coinciding with *DE*, *AC* will fall on *DF*, because the angle *BAC* is equal to the angle *EDF*. [*Hypothesis*.

Therefore also the point *C* will coincide with the point *F*, because *AC* is equal to *DF*. [*Hypothesis*.

But the point *B* was shewn to coincide with the point *E*, therefore the base *BC* will coincide with the base *EF*;

because, *B* coinciding with *E* and *C* with *F*, if the base *BC* does not coincide with the base *EF*, two straight lines will enclose a space; which is impossible. [*Axiom* 10.

Therefore the base *BC* coincides with the base *EF*, and is equal to it. [*Axiom* 8.

Therefore the whole triangle *ABC* coincides with the whole triangle *DEF*, and is equal to it. [*Axiom* 8.

And the other angles of the one coincide with the other angles of the other, and are equal to them, namely, the angle *ABC* to the angle *DEF*, and the angle *ACB* to the angle *DFE*.

Wherefore, *if two triangles* &c. q.e.d.

PROPOSITION 5. *THEOREM*.

The angles at the base of an isosceles triangle are equal to one another; and if the equal sides be produced the angles on the other side of the base shall be equal to one another.

Let *ABC* be an isosceles triangle, having the side *AB* equal to the side *AC*, and let the straight lines *AB*, *AC* be produced to *D* and *E*: the angle *ABC* shall be equal to the angle *ACB*, and the angle *CBD* to the angle *BCE*.

In *BD* take any point *F*,

and from *AE* the greater cut off *AG* equal to AF the less, [I.3.