We have to draw an angle of 135° using ruler and compasses only

We follow the following algorithm for the construction

Steps of construction

STEP1: Draw a line PC and take a point B on it.

STEP2: With B as a centre, and taking convenient radius, draw an arc, intersecting the ray BC at point N.

STEP3: With N as a centre, and taking the same radius, draw an arc cutting the previous arc at M.

STEP4: With M as a centre, and the same radius, draw an arc cutting the arc drawn in STEP 2 at L.

STEP5: With M as a centre, and the same radius draw an arc.

STEP6: With L as a centre, and the same radius, draw an arc cutting the arc drawn in STEP 5 at A

STEP7: Draw the ray BA.

STEP8: Bisect the angle ABP. Let BR be the bisecting ray.

Thus angle RBC is the required angle of 135°