Construction of Angles: Using Protractor and Compass, Steps of Constructions, Examples

February 10, 2024

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Introduction Construction of Angles

The construction of angles is one of the crucial dimensions of geometry. It is the pure form of geometry. Learning the construction of angles helps to deal with many other disciplines of mathematics and other domains. The term “construction” includes drawing and creating various shapes. To construct angles we need a pencil, a ruler and a pair of compasses or a protector.

Types of Angles

Acute Angle: Angles that measure less than 90 °s for example 60 °.
Obtuse angle: Angles which measure more than 90 °s but less than 180 °s for example 120 ° angle.
Straight angle: Angles equal to 180-° angle. A straight angle is a straight line.
Reflex angle: Angles which measure more than 180 °s but less than the 360-° angle for example 270 ° angle.
Complete angle: Angle equal to 360 °.

Angle Construction Using the Protractor

Protractor: It is a tool for calculating or measuring angles of a specific size. As illustrated in the following diagram, it is a semicircular portion of a geometric instrument made of metal or plastic that is marked in degrees from 0° to 180° both from left to right and from right to left:

Steps to construct angle using a protractor

Step 1: Draw a straight line AB using a pencil and ruler.
Step 2: Put the centre point of the straight part of the protractor on point A so that 0°s lie on point B.
Step 3: Mark a point C at 60 °s starting from B.
Step 4: Join points A and C.

Hence, ∠CAB is the required angle.

Using a Compass and Ruler to Create Angles

1. Constructing acute angles

Steps to construct a 60° angle

Step 1: Draw a straight line segment XY
Step 2: Set the compass in the length of the line segment.
Step 3: Draw an arc keeping the pointer of the compass on point Y i.e Y as the centre
Step 4: Draw another arc from the point where the first arc intersects the line segment XY. Name this point of intersection of the two arcs as Z
Step 5: Join points X and Z.

The angle ZYX so formed is 60° in measure

Note: Don’t change the setting of the compass

Steps to Construct 30°

30° angle can be formed by bisecting a 60° angle. Therefore to form a 30° angle first we need to construct a 60° angle

Note: You may follow the steps mentioned above to construct a 60° angle

Bisecting the angle ZYX

Step 1: Draw an arc X as the centre intersecting the arm XZ and XY.
Step 2: Name the new point of intersection as O and P.
Step 3: Draw two arcs keeping O and P as the centre respectively intersecting each other.
Step 4: Name the resultant point of intersection as Q.
Step 5: Joinpoint X and point Q

The angle QXY so formed is 30° to measure.

2. Constructing a Right Angle(90°)

Step 1: Draw a line segment ST.
Step 2: Set the compass about three-fourths of the length of line segment ST.
Step 3: Draw an arc keeping S as the centre.
Step 4: Draw another arc keeping T as the centre intersecting the first arc.
Step 5: Draw a straight line PR passing through the two points of intersection.

Constructing a 45° angle

An angle of 45° can be formed by bisecting the angle of 90°.

3. Constructing an obtuse angles

Steps to construct 120° angle

Step 1: Draw a ray EX.
Step 2: Taking E as the centre draw an arc of any radius, and draw an arc cutting EX at T.
Step 3: Draw another arc V taking T as the centre keeping the radius unchanged.
Step 4: Draw an arc U taking V as the centre keeping the radius the same.
Step 5: Join E and U.

The form angle UEX is 120° angle

Constructing an angle of 135°

Step 1: Draw a line segment AOB.
Step 2: Taking O as the centre draw an arc intersecting AOB at C.
Step 3: Draw another arc keeping C as the centre intersecting the arc formed before at D.
Step 4: Taking C and D as the centre draw two arcs intersecting each other.
Step 5: Name the new point of intersection as E.
Step 6: Draw a line segment passing through points E and point O.
Step7: Bisect the angle EOA and name the point of bisection at F.
Step 8: Joinpoint F and point O.

The angle so formed FOB is 135°

4. Constructing a straight angle (180°)

Step 1: Draw a line segment LM.
Step 2: Taking O as the centre anywhere at LM, draw an arc with a compass intersecting the line to the left of O and right of O.
Step 3: Name the two points of intersection as point N and point P.

The angle NOP is 180°

Construction of Angles FAQs

Q1. What are supplementary angles?

Ans: Supplementary angles are the pair of angles, the sum of which is equal to 180. Two supplementary angles form straight angles

Q2. In geometry, how is an angle made?

Ans: In Euclidean geometry, an angle is a figure that is created by two rays that share a terminus and are referred to as the angle’s sides and vertices. Two rays that are in the same plane as each other are used to create angles. Angles can also be created when two planes intersect.