How can we construct an angle using compass and ruler ??

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Constructing Angles of 60º, 120º, 30º and 90ºIn this section, we will consider the construction of some angles with special sizes.Constructing a 60º Angle

We know that the angles in anequilateral triangleare 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 armPQ.Step 2: Place the point of thecompassatPand draw anarcthat passes throughQ.Step 3: Place the point of the compass atQand draw an arc that passes throughP. Let this arc cut the arc drawn in Step 2 atR.

Constructing a 30º Angle

We know that:

So, to construct an angle of 30º, first construct a 60º angle and thenbisectit. Often, we apply the following steps.

Step 1: Draw the armPQ.Step 2: Place the point of thecompassatPand draw anarcthat passes throughQ.Step 3: Place the point of the compass atQand draw an arc that cuts the arc drawn in Step 2 atR.Step 4: With the point of the compass still atQ, draw an arc nearTas shown.Step 5: With the point of the compass atR, draw an arc to cut the arc drawn in Step 4 atT.Step 6: JoinTtoP. The angleQPTis 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 armPA.Step 2: Place the point of thecompassatPand draw anarcthat cuts the arm atQ.Step 3: Place the point of the compass atQand draw an arc ofradiusPQthat cuts the arc drawn in Step 2 atR.Step 4: With the point of the compass atR, draw an arc of radiusPQto cut the arc drawn in Step 2 atS.Step 5: With the point of the compass still atR, draw another arc of radiusPQnearTas shown.Step 6:With the point of the compass atS, draw an arc of radiusPQto cut the arc drawn in step 5 atT.Step 7: JoinTtoP. The angleAPTis 90º.