Constructing Angles

As we know that an angle is the coming together of two rays at one common point. The two rays are known as the arms and the common point is known as the vertex. Here, in this article, we will discuss the different types of angles and the step-by-step procedure to construct any angle. The angles can be categorized into different types like the acute angle, obtuse angle, right angle, etc.

Categorization of Angles

First, we will categorize angles on the basis of their measure or the degree of the angle. Now, we will come to know that with the increase in the measure of the angle, The Angle starts being known by different names. Now we will see what these names are with the pictorial representation of The Angles.

Acute Angle: – An acute angle is an angle that has a measure that is less than 90°. An example of an acute angle is an angle of 64°

Right Angle: – A right angle is an angle whose measure is exactly equal to 90°. 

Obtuse Angle: – An obtuse angle is an angle that has a measure that is greater than 90°. An example of an obtuse angle is 108°. 

Straight Angle: – A straight angle is an angle that has a measure of exactly 180°. 

Reflex Angle: – A Reflex Angle is an angle that has a measure of greater than 180°.  

Full Rotation Angle: – A Full Rotation Angle is an angle that has a measure of exactly 360°. 

 Now we will categorize the angles on the basis of their direction of rotation. This category comprises two types of angles: – Positive Angle and Negative Angle.

Positive Angle: – An Angle that is measured in a direction of anti-clockwise from the baseline is called a Positive Angle. 

Negative Angle: – Similarly a Negative Angle is one that is measured in the direction clockwise from the baseline. 

Construction of Angles

In this topic, we will construct various angles of measures like 30° and 60°. First, we will start with the construction of an angle of the measure of 60°.

Construction of angle of  60°

A triangle that has all the three angles equal to 60°. So we should work along the lines of the concept of the equilateral triangle. So first we have to draw sort of an equilateral triangle. Let me explain how.

First, draw a line segment. Name the two ends of the Line Segment as P and Q.

Now take your compass, and place the pointy end of the compass at P. From there, draw an arc that passes through Q. Now place the pointy end of the compass at Q, draw an arc that passes through P. Mark the meeting point of both the arcs as R. Join P with R. Now we have our angle. ∆QPR is our required angle of 60°.

Construction of angle of 30°

Now we will see how to construct an angle of 30° by using the concept of the construction of an angle of 60°.

Since we know that the half of 60 is 30. So first we have to construct an angle of 60° and then bisect it.

So first we draw a line segment. Name the two ends of the Line Segment as P and Q.

Now take your compass, and place the pointy end of the compass at P. From there, draw an arc that passes through Q. Now place the pointy end of the compass at Q, draw an arc that passes through P. Mark the meeting point of both the arcs as R. Now without removing the pointy end of the compass from Q, draw the arc towards the right side. Now place the pointy end of the compass at R, draw an arc that bisects the arc drawn from Q. Name the meeting point of these two arcs as T. Join P with T. Now we have our ∆QPT as 30°.

Solved Problems

Q. Construct an angle of 90°.

Soln.

For constructing an angle of 90°, follow the steps given below.

First, draw a Line Segment. Name the two ends of the Line Segment as P and A. 

Place the pointy end of the compass at P. Draw an arc that bisects the Line Segment. Let the point at which the arc meets the Line Segment be Q. Now take the radius as the length PQ. Place the pointy end of the compass at Q and draw an arc that bisects the arc drawn from P. Let the point of intersection of the two arcs be R. Now take the same radius PQ, place the pointy end of the compass at R and draw an arc which bisects the arc drawn from P. Let the point of intersection of these two arcs be S. Now take the same radius PQ, place the pointy end of the compass at R and draw an arc at the top. Now take the same radius PQ, place the pointy end of the compass at S and draw an arc that meets the last arc. Let the point of the intersection of these two arcs be T. Now we have our ∆APT as 90°.

Conclusion

In this article, we saw the categorization of angles, construction of angles, and solved problems related to the construction of angles. We know that angle is the geometric figure formed by the coming together of two rays at a particular point. We have discussed the step-by-step procedure of constructing angles. The construction of angle is a very important aspect of geometry.