Year 8 Interactive Maths - Second Edition

## Constructing Angles of 60º, 120º, 30º and 90º

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. #### Example 12 ##### Solution:    ### Activity 10.3 