For a better understanding of the solution and the explanation given here please go through the diagram in the file attached.
By definition, measure of an arc is the angle subtended by the arc at the centre of the circle. Thus, the measure of the arc [tex] \overarc {RSQ} [/tex] is the measure of the [tex] \angle RAQ [/tex] (in red) which the arc [tex] \overarc {RSQ} [/tex] subtends at the center.
Now, in order to find the measure of the [tex] \angle RAQ [/tex] (in red) we will have to use the information given in the question. We have been told that [tex] m\angle RSQ=24^{\circ} [/tex]. Now, we know that the double angle theorem says that, the angle subtended at the centre of a circle is double the size of the angle subtended at the circumference from the same two points. Therefore, applying this theorem we get:
[tex] \angle RAQ =2\times 24^{\circ}=48^{\circ} [/tex]
Thus if [tex] \angle RAQ [/tex] (in red)=[tex] 48^{\circ} [/tex], then [tex] \angle RAQ [/tex] (in green) will be equal to measure of the arc [tex] \overarc {RSQ} [/tex]. Now, [tex] \angle RAQ [/tex] (in green) will obviously be:
[tex] \angle RAQ [/tex] (in green)=[tex] 360^{\circ}-\angle RAQ [/tex] (in red)
[tex] \angle RAQ [/tex] (in green)=[tex] 360^{\circ}-48^{\circ} [/tex]
[tex] \angle RAQ [/tex] (in green)=[tex] 312^{\circ} [/tex]
Thus, the measure of arc [tex] \overarc {RSQ} [/tex]=[tex] 312^{\circ} [/tex]
