Construction of Angles(Using Protractor & Compass)

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An angle is a shape formed by two rays that share a common point (called a vertex). The construction of angles is one of the essential part of geometry. We can use a protractor to construct any type of angle. Also, there are methods by which we can construct some specific angles such as $60^{\circ}$, $30^{\circ}$, $120^{\circ}$, $90^{\circ}$, $45^{\circ}$, etc., using a compass and ruler (without using a protractor).

Let’s understand the procedure of construction of angles.

Construction of Angles Using a Protractor

With the help of a protractor, you can construct an angle of any measure. 

The steps involved in the construction of angles using a protractor are as follows.

Step 1: Draw a line

Step 2: Mark two points $\text{A}$ and $\text{B}$ on it.

Step 3: Place the centre of the protractor on point $\text{A}$, such that the line segment $\text{AB}$ is aligned with the line of the protractor.

Step 4: Starting from $0$ (in the protractor) mark the point $\text{C}$ on the paper as per the required angle.

Step 5: Join points $\text{A}$ and $\text{C}$. $\angle \text{BAC}$ is the required angle.

Step 6: Extend the line segment $\text{AC}$ as required.

Construction of Angles Using a Compass and Ruler

The two basic constructions using a compass and ruler are

• construction of 60 degree angle
• bisecting an angle

If you know these two constructions, you can construct angles like $30^{\circ}$, $15^{\circ}$, $45^{\circ}$, $90^{\circ}$, $22.5^{\circ}$, $120^{\circ}$, $135^{\circ}$, etc.

Construction of 60 Degree Angle

The steps involved in the construction of 60 degree angle are

Step 1: Draw a line segment. Mark the left end as point $\text{O}$ and the right end as point $\text{B}$.

Step 2: Take the compass and open it up to a convenient radius. Place its pointer at $\text{O}$ and with the pencil head make an arc that meets the line $\text{OB}$ say at $\text{P}$.

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

Step 4: Draw a line from $\text{O}$ through $\text{A}$.

We get the required angle i.e. $\angle \text{AOB} = 60^{\circ}$.

Bisecting an Angle

Let’s start with an $\angle \text{AOB}$.

The steps involved in the bisection of an angle are

Step 1: Take the compass and open it up to a convenient radius. With $\text{O}$ as the centre, draw two arcs such that it cut the rays $\text{OA}$  and $\text{OB}$ at points $\text{C}$ and $\text{D}$ respectively.

Step 2: Without changing the distance between the legs of the compass, draw two arcs with $\text{C}$ and $\text{D}$ as centres, such that these two arcs intersect at a point say $\text{E}$.

Step 3: Join the $\text{O}$ with $\text{E}$.

$\text{OE}$ is the required angle bisector of $\angle \text{AOB}$.

Construction of Special Angles

As mentioned above, you can construct some of the special angles, if you know the construction of a $60^{\circ}$ angle and the bisection of an angle.

Here are the steps for constructing some of the angles.

Construction of 120 Degree Angle

A $120^{\circ}$ angle is exactly double that of A $60^{\circ}$ angle. If we know the construction of a $60^{\circ}$ angle, then we can easily construct a $120^{\circ}$ angle.

Here are the steps to construct a $120^{\circ}$ angle.

Step 1: Draw a line segment $\text{AB}$.

Step 2: With $\text{A}$ as a centre and draw an arc of proper length.

Step 3: Take $\text{D}$ as a centre with the same radius, and draw two marks $\text{E}$ and $\text{F}$ on the former arc.

Step 4: Join points $\text{A}$ and $\text{F}$ and produce to point $\text{C}$. Thus $\angle \text{CAB} = 120^{\circ}$.

Construction of 90 Degree Angle

A $90^{\circ}$ angle lies exactly between a $60^{\circ}$ angle and a $120^{\circ}$ angle. If we know the construction of $60^{\circ}$ and $120^{\circ}$ angles, then we can easily construct $90^{\circ}$ angle. 

Here are the steps to construct a $90^{\circ}$ angle.

Step 1: Draw a line segment $\text{OA}$.

Step 2: Taking $\text{O}$ as centre and using a compass draw an arc of some radius, that cuts $\text{OA}$ at $\text{B}$.

Step 3: Taking $\text{B}$ as centre and with the same radius draw another arc, that cuts the first arc at $\text{C}$.

Step 4: Taking $\text{C}$ as centre and with the same radius draw an arc, that cuts the first arc at $\text{D}$.

Step 5: Now taking $\text{C}$ and $\text{D}$ as centres and radius greater than the arc $\text{CD}$, draw two arcs, such that they intersect at $\text{E}$.

Step 6: Join $\text{OE}$ such that $\angle \text{AOE}$ is a $90^{\circ}$ angle.

Construction of 45 Degree Angle

A $45^{\circ}$ angle is exactly half of $90^{\circ}$. If we know the construction of $90^{\circ}$ angle and bisector of an angle, we can easily construct a $45^{\circ}$.

Here are the steps to construct a $45^{\circ}$ angle.

Step 1: Draw a line segment $\text{AB}$ on a plane sheet.

Step 2: With the centre, $\text{B}$ draw an arc that meets $\text{AB}$ at $\text{C}$.

Step 3: Take $\text{C}$ as a centre and with the same radius, mark two small arcs $\text{D}$ and $\text{E}$ on the former arc.

Step 4: Take $\text{D}$ and $\text{E}$ as centres and with the same radius, draw two arcs that meet each other at point $\text{G}$.

Step 5: Join points $\text{B}$ and $\text{G}$ such that $\angle \text{ABG} = 90^{\circ}$

Step 6: Draw the angle bisector $\text{BH}$ of $\angle \text{ABG}$ such that $\angle \text{ABH} = 45^{\circ}$.

Practice Problems

1. Name the instrument with which you can construct an angle of any measure.
2. What are the two basic constructions required to construct angles using a ruler and compass?
3. Write down the steps to construct the following angles
   • $30^{\circ}$
   • $15^{\circ}$
   • $22.5^{\circ}$
   • $45^{\circ}$
   • $7.5^{\circ}$

FAQs

Conclusion

The construction of angles is one of the essential part of geometry. You can use a protractor to construct any type of angle. You can also construct certain specific angles using a ruler and a compass.

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