The sum of angles at a point (or in a circle) is 360 degrees.
Therefor, we have that:
[tex]m\angle GCD+m\operatorname{\angle}DCE+m\operatorname{\angle}ECF+m\operatorname{\angle}FCG=360\degree[/tex]
Given the values in the question:
[tex]\begin{gathered} m\operatorname{\angle}DCE=75 \\ m\operatorname{\angle}ECF=80 \\ m\operatorname{\angle}FCG=55 \end{gathered}[/tex]
Therefor, we can calculatoe the measure of angle GCD to be:
[tex]\begin{gathered} m\operatorname{\angle}GCD+75+80+55=360 \\ m\operatorname{\angle}GCD=360-75-80-55 \\ m\operatorname{\angle}GCD=150\degree \end{gathered}[/tex]
