Draw an angle of 45° using scale and compasses only. Draw the bisector of this angle.

Steps of construction:
1. Draw a line segment AB.
2. Draw an angle $\angle$BAC = 90$°$.
3. With A as the centre and a small radius, draw an arc cutting AB at P and AC at T.
4. With P as the centre and a radius more than half of PT, draw an arc.
5. With T as the centre and the same radius as above, draw another arc cutting the previously drawn arc at N .
6. Join AN cutting the arc at Z and produced it to X to make the ray AX.
7. With P as the centre and a radius more than half of PZ, draw an arc.
8. With Z as the centre and the same radius as above, draw another arc cutting the previously drawn arc at M .
9. Join AM.
Thus, AM bisects $\angle$BAX.