The angle between the external bisectors of two angles of a triangle is $60^{\circ }$, then the third angle of the triangle is

The angle formed between the external bisectors of two angles of a triangle is mathematically related to the internal third vertex angle ($A$) by the theorem:
$$\theta =90^{\circ }-\frac{A}{2}$$

We are given that the angle between the external bisectors is $\theta =60^{\circ }$:

$$60^{\circ }=90^{\circ }-\frac{A}{2}$$

$$\frac{A}{2}=90^{\circ }-60^{\circ }=30^{\circ }$$

$$A=30^{\circ }\times 2=60^{\circ }$$