1. The problem states that the angles of a triangle are in the ratio 5 : 6 : 7. 2. Let the common ratio factor be $x$. Then the angles can be expressed as $5x$, $6x$, and $7x$. 3. The sum of the angles in any triangle is always $180^\circ$. 4. Therefore, we write the equation: $$5x + 6x + 7x = 180$$ 5. Simplify the left side: $$18x = 180$$ 6. Solve for $x$: $$x = \frac{180}{18} = 10$$ 7. The smallest angle corresponds to the smallest ratio, which is $5x$. 8. Calculate the smallest angle: $$5 \times 10 = 50^\circ$$ The size of the smallest angle is $50^\circ$.