Construction of Angles Using Protractor and Compass

Constructing an angle is the most basic and important skill in geometry, but what is an angle? An angle is what we get when we combine two rays at a common endpoint. There are 2 ways of constructing angles, we can either use a protractor to construct an angle or we could use instruments such as a ruler and compass. Here we shall learn the different types of angles and how to construct angles such as 60°, 120°, 90°, 45°, 30° etc. Using a protractor, compass, and rulers. 

Read Also: Pair of Angles

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Types of Angles

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There are six types of angles and they can be distinguished based on the degree that they form. 

Acute Angle: - An angle whose measure is less than 90°. 

Right Angle: - An angle whose measure is exactly 90°. 

Straight Angle: - An angle whose measure is equal to 180° and this angle looks like a straight line. 

Obtuse Angle: - An angle whose measure is greater than 90° but less than 180°. 

Reflex Angle: - An angle whose measure is always more than 180° but less than 360°. 

Full Rotation Angle: - An angle whose measure is exactly 360°. 

Types of Angles

Read More: Circumference of a circle

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Constructing an Angle Using a Protractor

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It is really simple to construct an angle by using a protractor. We shall use the protractor to mark the measure of the angle and then construct it using a pencil and ruler.

Step 1: First we shall draw a line segment CD. 

Step 2: Now we must place the center of the protractor on point C, such that the line segment CD is aligned with the line of the protractor. 

Step 3: Starting from the 0° mark on the protractor we shall mark the required measure of the angle as point E. 

Step 4: Now join the points C and E. Hence, we form ∠ECD which is the required angle. 

Constructing an Angle Using a Protractor

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Constructing an Angle Using a Ruler and a Compass

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We shall now learn how to construct an angle using a compass and ruler. Using this method only a few particular angles can be constructed such as 60°, 120°, 90°, 45°, 30°.

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Construction of a 60° Angle

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The 60 degrees angle is one of the most basic and easiest constructions, using this angle we can construct various other angles. 

Here are the steps to be followed while constructing a 60° angle: - 

Step 1: First we shall draw a line segment AI. 

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

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

Step 4: Draw a line from A through P. 

60° Angle

Hence the required angle is ∠BOP = 60-degree angle. 

Check Also: Decimal Expansion

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Construction of a 30° Angle

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A 30° angle is half of a 60 degrees angle is hence made by bisecting a 60 degrees angle. Here are the steps to be followed: - 

Step 1: - First we shall follow the above-mentioned steps and create a 60-degree angle. 

Step 2: - Now we are required to bisect the angle. We shall bisect the angle by taking the radius as the length between points A and B. Now place the pointer of the compass at point B and make an arc and similarly make another arc by placing the pointer at point P. 

Step 3: - Draw a line through the intersection of the two arcs and name it R.

30° Angle

Hence the required angle is ∠RAB = 30°. 

Check Important Notes for Alternate Angles

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Construction of a 120° Angle

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Step 1: Lets first draw a line AB. On line AB take a point O. 

Step 2: Now we shall take the compass and open it up to any radius and place the pointer of the compass at point O on line AB. With the pencil-head of the compass make an arc which meets the line AB at say a point, P. 

Step 3: Now without changing the radius, place the pointer at point P and make an arc that cuts the previous arc at a point, say Q. 

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

Step 5: Draw a line from O through R. 

120° Angle

Hence, ∠ROB is the required 120 degrees angle. 

Read Further: Half Angle Formula

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Construction of a 90° Angle

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A 90-degree angle is easy to make as it lies exactly between a 120-degree angle and a 60-degree angle and we have already learned how to construct these angles. 

Here are the steps to be followed: - 

Step 1: - First we shall draw a line m. On line m we shall take a point D. 

Step 2: - Taking D as the center and we use a compass to draw an arc of any radius to cut line m at a point say, S. 

Step 3: - Taking S as the center and with the same radius that we had taken previously, draw another arc, that cuts the first arc at P. 

Step 4: - Taking P as the center and with the same radius draw an arc, that cuts the first arc at a point say, W. 

Step 5: - Now taking W and P as centers and with a radius greater than that of arc WS, draw two arcs, such that they intersect at point say, O. 

Step 6: - Join OD such that ∠ODS is a 90-degree angle

90° Angle

Hence the required angle is ∠ODS = 90°angle

Check More: Types of Triangles

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Construction of a 45° Angle

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The 45° angle is half of the 90° angle and can be constructed by bisecting the 90° angle. 

Here are the steps to construct it: - 

Step 1: - First construct the 90-degree angle as shown previously. 

Step 2: - Now we shall bisect the angle by taking the points S and Z as the centers and drawing two arcs that intersect at a point say, X. 

Step 3: - Draw a line through X from point D 

45° Angle

Hence the required angle is ∠XDS = 45° angle 

Read Also: Lines and Angles

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Things to Remember

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1. Constructing an angle is the most basic and important skill in geometry. An angle is what we get when we combine two rays at a common endpoint.
2. A ray that divides an angle into two equal parts is known as an angle bisector. 
3. Always remember to measure the angle with a protractor after you have constructed it with a ruler and compass. 
4. With the help of a ruler and a compass, we can only construct the angels, 90°,60°,45°,22.5°,30° etc., and its bisecting of an angle. 
5. At the end of the construction of the angle always mention the required angle. 
6. Do not erase the arcs or constructed lines after you have finished construction. 

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