Construct an angle of $135^{\circ}$ and verify it by measuring with a protractor.

(N/A) Steps of construction:
$I.$ Draw a ray $\overrightarrow{OP}$.
$II.$ With centre $O$ and a suitable radius,draw an arc to meet $\overrightarrow{OP}$ at $A$.
$III.$ Keeping the same radius and starting from $A$,mark points $Q, R,$ and $S$ on the arc such that $\angle AOQ = 60^{\circ}$,$\angle AOR = 120^{\circ}$,and $\angle AOS = 180^{\circ}$.
$IV.$ Draw $\overrightarrow{OL}$,the bisector of $\angle ROS$,which makes $\angle ROL = 30^{\circ}$. Thus,$\angle AOL = 120^{\circ} + 30^{\circ} = 150^{\circ}$.
$V.$ Draw $\overrightarrow{OM}$,the bisector of $\angle ROL$,which makes $\angle ROM = 15^{\circ}$. Thus,$\angle AOM = 120^{\circ} + 15^{\circ} = 135^{\circ}$.
Therefore,$\angle AOM = 135^{\circ}$.