Whenever a petition as is now provided by law is presented to the county commission of any county for the opening, extension or widening of any county road along and adjoining the corporate limits of any city of the third class, such commission shall proceed as is now authorized by law to condemn the right-of-way of such road, the maximum width of which shall not be over eighty feet. Before such road, however, shall be declared opened or the possession of such property for such right-of-way be taken for public use, the council of such city shall ascertain, in the manner herein provided in this chapter, the proportionate amount of benefits resulting to the owners of land adjacent thereto within the corporate limits of such city, who shall contribute toward compensating the person injured, and such amount shall constitute a lien in favor of the city on the said adjacent land, and the person or persons who shall be benefitted and so assessed shall pay in such manner as herein provided. The city authorities shall define by ordinance the limits within which private property is deemed benefitted by the opening, extension or widening as aforesaid, and the owners of private property within such limits shall be notified as herein set out, and appeals may be taken as herein provided.