Construct an angle of the measure ${45}^{\circ }$ with ruler and compasses.

Steps to construct an angle of measure ${45}^{\circ }:$

Step 1: Draw a ray $OA$

Step 2: Taking $O$ as the center and any radius, draw an arc cutting $OA$ at $B$
Step 3: With $B$ as centre and with the same radius as before, construct an arc intersecting the previous arc at $C.$

Step 4: With $C$ as centre and with the same radius, draw an arc intersecting the arc at $D.$

Step 5: With $C$ and $D$ as centres and radius more than $\frac{1}{2}CD,$ draw two arcs intersecting at $P.$

Step 6: Join $OP.$

$\therefore \angle AOP={90}^{\circ }$
Step 7: Let $OP$ intersect the original arc at $Q$
Step 8: With $B$ and $Q$ as centres and radius greater than $\frac{1}{2}BQ,$ draw two arcs intersecting at $R$
Step 9: Join $OR$
Thus $\angle AOR={45}^{\circ }$