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
Key Terms
constructing angles |