Draw an angle of measure 127° and construct its bisector.

Steps:

1. Draw a ray OA.

2. Place the centre of the protractor at O and the zero line along OA. Mark a point at 127°.

3. Remove the protractor and join O with the marked point and produce it to B. Then ∠AOB = 127°.

4. With O as centre and any (suitable) radius, draw an arc to meet ray OA at C and ray OB at D.

5. With C as centre and any suitable radius $\left(\gt\dfrac{1}{2}\text{CD}\right)$, draw an arc. Also, with D as centre and same radius, draw another arc to meet the previous arc at E.

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

Hence, ray OE is the required bisector of ∠AOB, and ∠AOE = ∠EOB = 63.5°.