Constructing Angles of 60º, 120º, 30º and 90ºwith the compass

In this section, we will consider the construction of some angles with special sizes. Constructing a 60º Angle We know that the angles in an equilateral triangle are all 60º in size.  This suggests that to construct a 60º angle we need to construct an equilateral triangle as described below. Step 1:  Draw the arm PQ. Step 2:  Place the point of the compass at P and draw an arc that passes through Q. Step 3:  Place the point of the compass at Q and draw an arc that passes through P.  Let this arc cut the arc drawn in Step 2 at R. Constructing a 30º Angle We know that: So, to construct an angle of 30º, first construct a 60º angle and then bisect it. Often, we apply the following steps. Step 1:  Draw the arm PQ. Step 2:  Place the point of the compass at P and draw an arc that passes through Q. Step 3:  Place the point of the compass at Q and draw an arc that cuts the arc drawn in Step 2 at R. Step 4:  With the point of the compass still at Q, draw an arc near T as shown. Step 5:  With the point of the compass at R, draw an arc to cut the arc drawn in Step 4 at T. Step 6:  Join T to P.  The angle QPT is 30º. Constructing a 120º Angle We know that: This means that 120º is the supplement of 60º.  Therefore, to construct a 120º angle, construct a 60º angle and then extend one of its arms as shown below. Constructing a 90º Angle We can construct a 90º angle either by bisecting a straight angle or using the following steps. Step 1:  Draw the arm PA. Step 2:  Place the point of the compass at P and draw an arc that cuts the arm at Q. Step 3:  Place the point of the compass at Q and draw an arc of radius PQ that cuts the arc drawn in Step 2 at R. Step 4:  With the point of the compass at R, draw an arc of radius PQ to cut the arc drawn in Step 2 at S. Step 5:  With the point of the compass still at R, draw another arc of radius PQ near T as shown. Step 6:  With the point of the compass at S, draw an arc of radius PQ to cut the arc drawn in step 5 at T. Step 7:  Join T to P. The angle APT is 90º. Please print this worksheet to practise the angles construction with the compass: