Draw an angle of measure 45° and bisect it.

Steps:

1. Draw any line l and take a point O on it.

2. With O as centre and suitable radius, draw two arcs to cut line l at points A and B.

3. With A and B as centres, draw two arcs of equal radius $\left(\gt\dfrac{1}{2}\text{AB}\right)$ cutting each other at C. Join OC and produce it. Then ∠BOC = 90°.

4. Now we bisect ∠BOC by drawing two equal arc from point B and F, cutting each other at point D. Join OD and the bisector is ray OD. Then ∠BOD = 45°.

5. Now to bisect ∠BOD, with O as centre and any suitable radius, draw an arc to meet OB at P and OD at Q.

6. With P and Q as centres and equal radii $\left(\gt\dfrac{1}{2}\text{PQ}\right)$, draw two arcs cutting each other at E.

7. Join OE and produce it to form a ray.

Hence, ray OE is the bisector of ∠BOD = 45°, and ∠BOE = ∠EOD = 22.5°.