Construction Of Special Angles

Before talking about constructing angles, let us quickly recall the types of angles. Depending on the inclination between the two arms, an angle may be acute (less than 90 degrees, say 60 degree angle), obtuse (more than 90 degrees) or right-angled (exactly 90 degrees). Constructing angles is an important part of geometry as this knowledge is extended for construction of other geometric figures as well, primarily the triangles.

In this article, we will discuss constructing angles of measures 60 degrees and 120 degrees. Using these you will also learn the 30 degree, 90 degree and 45 degree angle constructions using the geometric tools- a compass and a ruler.

Steps for Constructing Angles

60 degree angle

60 degree is one of the most basic constructions, which facilitates constructing angles of several other measures. The steps are:

  • Step 1: Draw a line segment. Mark the left end as point O and the right end as point B.

Angles - Construction Of Special Angles

  • Step 2: Take the compass and open it up to a convenient radius. Place its pointer at O and with the pencil-head make an arc which meets the line OB at say, P.

Angles - Construction Of Special Angles

  • Step 3: Place the compass pointer at P and mark an arc that passes through O and intersects the previous arc at a point, say A.

Angles - Construction Of Special Angles

  • Step 4: Draw a line from O through A.

Angles - Construction Of Special Angles

We get the required angle. ∠AOB = 60 degree angle.

30 degree angle

A 30 degree angle is the half of 60 degree angle. For its construction, you first construct a 60 degree angle as discussed above. Then, you bisect this angle. You get two 30 degree angles.

120 degree angle

A 120 degree angle is the double of 60 degree angle. The steps for its construction are:

  • Step 1: Draw a line segment. Mark the left end as point O and the right end as point B.

Angles - Construction Of Special Angles

  • Step 2: Take the compass and open it up to a convenient radius. Place its pointer at O and with the pencil-head make an arc which meets the line OB at say, P.

Angles - Construction Of Special Angles

  • Step 3: Without disturbing the radius, place the pointer at P and make an arc that cuts the previous arc at a point, say Q.

Angles - Construction Of Special Angles

  • Step 4: Similarly, with the same radius on the compass, place the pointer at point Q. Mark another arc on the first arc. Mark the point where they intersect as A.

Angles - Construction Of Special Angles

  • Step 5: Draw a line from O through A.

Angles - Construction Of Special Angles

We get the required angle, ∠AOB = 120 degrees.

90 degree angle

A 90 degree angle lies exactly half way between a 120 degree angle and a 60 degree angle on a 360 degree scale. For its construction, you first construct a 60 degree angle and a 120 degree angle as discussed above. Then, you construct a bisector between these two angles. You get the required 90 degree angle.

45 degree angle

A 45 degree angle is the half of 90 degree angle. For its construction, you first construct a 90 degree angle as discussed above. Then, you bisect this angle. You get two 45 degree angles.

For constructing angles like 15 degrees or 22.5 degrees, you further bisect 30 degree angle and 45 degree angle respectively.

From the above discussion , one would be able to understand the importance of special angles in the field of geometry. Know further about special angles by solving sample question and answers and to model questions on their own here at NCERT Solutions for Construction

To learn more about constructing angles of different measures, download Byju’s the Learning App.


Practise This Question

Draw a line AB. At A, draw an arc of length 3cm using compass such that it intersects AB at O. With the same spread of compass, put the compass pointer at O and make an arc that intersects the previous arc at P. With the same spread again, put the compass pointer at P and draw an arc that intersects the first arc at Q. Join A and Q. Using the protractor, measure QAB. What is the value of QAB.