how do we draw a triangle with two of the angels 40 degrees and 120 degrees and circumradius 3 centimeters ?

Answer 1

start with a sketch of the finished product
Let the triangle be ABC, so that angle B = 120°,
angle BCA = 40° and angle CAB = 180-40-120 = 20°

the centre of the triangle, O, lies on the right-bisectors of BC and AB, so that
CO = AO = BO = 3 cm

remember that the central angle of a chord is twice the angle subtended by the same chord at the circle
so angle BOA = 80° and angle COB = 40° making angle COA = 120°
also in triangle COA, which is isosceles,
angle ACO = angle COA = 30°
all angles can be found.

So I would start by drawing a circle with radius 3 cm
Pick any point C,
Using your protractor , draw angle OCB = 70°, where B is the point on the circle
Using your protractor, draw angle COA = 120°, with A on the circle.
Depending on your accuracy of the protractor and your skill, triangle ABC is your required triangle.

NOTE: since you are using actual measurements, this is not the traditional straight-edge and compass construction.