How do we draw 45 degree, 90 degree, 120 degree, 30 degree and 60 degree using ruler and a compass.

Follow the given steps to construct an angle of 90º.

Step1: Draw a line l and mark a point O on it.

Step2: With O as centre and any convenient radius, draw an arc intersecting l in P and Q.

Step3: Taking P as centre and any convenient radius, draw an arc.

Step4: Taking Q as cente and same radius taken in step3, draw an arc intersecting the previously drawn arc in (say)R.

Step5: Join OR.

Here, ∠ QOR = ∠ POR = 90º

Follow the given steps to construct an angle of 60º.

Step 1: Draw a ray AB. Draw an arc of convenient radius, while taking point A as centre. Let it intersect the ray AB at C.

Step 2: Taking C as centre and with the same radius as before, draw an arc intersecting the previously drawn arc at D.

Step 3: Join AD

Here, ∠ DAC = 60º

Follow the given steps to construct an angle of 120º.

Step 1:  Draw a ray PQ. Draw an arc of convenient radius, while taking point P as centre. Let it intersect the ray PQ at R.

Step 2:  Taking R as centre and with the same radius as before, draw an arc intersecting the previously drawn arc at S.

Step 3: Taking S as centre and with the same radius as before, draw an arc intersecting the arc at T

Step 4: Join PT

Here, ∠ TPR = 120º

Follow the given steps to construct an angle of 30º.

Step 1: Draw a ray AB. Draw an arc of convenient radius, while taking point A as centre. Let it intersect the ray AB at C.

Step 2: Taking C as centre and with the same radius as before, draw an arc intersecting the previously drawn arc at D.

Step 3: Join AD. Now, taking C  and D as centres, draw arcs with radius more than ½ CD. Let these intersect each other at E. 

Step 4: Join AE.

Here, ∠ DAE = ∠ BAE = 30º

Similarly, the bisector of 90º gives the angle measuring 45º.