The angles of a quadrilateral are in the ratio of 5: 8 : 10 : 13. The smallest of these angles is:

The ratio of the angles of the quadrilateral = 5: 8: 10: 13

Sum of the angles of a quadrilateral = 360^∘

Let the angles of the quadrilateral be 5x, 8x, 10x, and 13x, where x is a constant.

Sum of the angles of the quadrilateral:

​$$5x + 8x + 10x + 13x = 360^\circ$$​​

​$$36x = 360^\circ$$​​

​$$x = \frac{360^\circ}{36} = 10^\circ$$​​

The angles of the quadrilateral are:

First angle = $$5x = 5 \times 10^\circ = 50^\circ$$​

Second angle = $$8x = 8 \times 10^\circ = 80^\circ$$​

Third angle = $$10x = 10 \times 10^\circ = 100^\circ$$​

Fourth angle = $$13x = 13 \times 10^\circ = 130^\circ$$​

The smallest angle is: $$50^\circ$$​

Thus, the smallest angle of the quadrilateral is $$50^\circ.$$