Two regular polygons are such that the ratio between their number of sides is 1 : 2 and the ratio of measures of their interior angles is 3 : 4. Then the number of sides of each polygon are

Each interior angle of polygon is given by
$$\eqalign{ & = \frac{{\left( {x - 2} \right)}}{x} \times 180 \cr & {\text{Sides is }}a,\,2a \cr & \frac{{\left( {\frac{{a - 2}}{a}} \right) \times 180}}{{\left( {\frac{{2a - 2}}{{2a}}} \right) \times 180}} = \frac{3}{4} \cr & \frac{{\left( {a - 2} \right)}}{a} \times \frac{{2a}}{{2\left( {a - 1} \right)}} = \frac{3}{4} \cr & \frac{{\left( {a - 2} \right)}}{{\left( {a - 1} \right)}} = \frac{3}{4} \cr & 4a - 8 = 3a - 3 \cr & a = 5 \cr & {\text{So, sides }}5,\,10 \cr} $$