By using a ruler and compass, construct an angle of 135° and bisect it. Measure any one part by protractor and see how accurate you are.

Steps:

1. Draw a straight line AOA'.

2. Construct a perpendicular OC to line AOA' at O. Then ∠AOC = 90°.

3. Bisect ∠A'OC by drawing two equal arcs from centre O, cutting line OA' at point D and OC at point E. Now, from points D and E draw two arcs of equal length $\left(\gt\dfrac{1}{2} \text{ED}\right)$ intersecting each other at point B. Join OB and OB is the bisector of ∠A'OC. Then ∠COB = 45°.

4. Therefore, ∠AOB = ∠AOC + ∠COB = 90° + 45° = 135°.

5. Now bisect ∠AOB. With O as centre, draw two equal arcs that cut OA at point P and OB at point 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 R.

7. Join OR and produce it to form ray OR.

Then OR is the bisector of ∠AOB.

On measuring one of the parts with a protractor, we find:

∠AOR = 67.5° (and ∠ROB = 67.5°).

Hence, ray OR bisects ∠AOB into two equal parts of 67.5° each.