Step 1: Picture the molecule.
Hydrogen peroxide $H_2O_2$ is not flat. It has an open-book shape, with the two O-H bonds on different pages of the book.
Step 2: Understand dihedral angle.
The dihedral angle is the angle between the two pages of the open book, that is between the plane of one $H-O-O$ part and the plane of the other $O-O-H$ part.
Step 3: Know that the shape changes with state.
The angle is different in the gas phase and in the solid phase, because in the solid the molecules pack together and hydrogen bonds squeeze the shape.
Step 4: Recall the gas phase value.
In the gas phase the molecules are free, so the shape opens up wider to reduce repulsion. The dihedral angle here is 111.5 degrees.
Step 5: Recall the solid phase value.
In the solid phase the hydrogen bonding pulls the molecule into a tighter shape and the dihedral angle drops to about 90.2 degrees.
Step 6: Match to the question.
The question asks for the gas phase, so the answer is the larger value. \[ \boxed{111.5^{\circ}} \]