Practical Geometry

Construction Related to Angles

If we look around we will see angles everywhere. While playing basketball if you want to shoot the ball in the basket, you must throw the ball at a particular angle. Also when a plane is about to land, the pilot calculates the angle in which the plane lands safely. So let us explore more about angles and angle bisector further.

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When two rays hold a common endpoint, in this case, the two rays together form an angle. Therefore, an angle is formed by two rays initiating from a shared endpoint. These two rays creating it are termed as the sides or arms of the angle. For representing an angle the symbol “∠” is used in geometry.


When the box is closed, the angle between them is zero degrees. When you try to open the box, you can see the angle is formed between the upper surface of the box and the bottom surface.

Constructing a Copy of an Angle


  1. Draw a line l with a ruler and with the help of a protractor, draw a ∠DAP of measure 70°
  2.  Place the compasses at A and draw an arc to cut the rays of A at B and C.
  3. Use the same compasses setting to draw an arc with P as the centre, cutting line L in Q.
  4. Set your compasses to the length BC with the same radius.
  5. Place the compasses pointer at Q and draw the arc to cut the arc drawn earlier in R.
  6.  Join PR. This gives the point P. It has the measure same as point A.

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Angle Bisector of a Given Angle

Angle Bisector

  1. Taking A as the centre and by using compasses, draw an arc that cuts both rays of A.  Also name the points of intersection as B and C.
  2. Now taking B as the centre, draw an arc whose radius is more than half the length BC.
  3. Again with the same radius and with C as the centre, draw another arc in the interior of A. Let the two arcs intersect at D. Then AD is the required angle bisector of A.

Drawing Angles of a Given Measure

1. Construction of 60° Angle

  1. Take a ruler and draw a line l and make a point O on it. Take a compass and put its one end at point O and draw an arc with any convenient radius.
  2. We take the centre O and draw an arc. We get the point A.
  3. Now take the compass and with the same orientation taking the point A as the centre draw an arc that passes through O. Draw an arc such that it intersects the existing arc.
  4. Here OA and OB are nothing but the radius which is equal in length.
  5. So actually we are trying to draw an equilateral triangle which means the angle we see in the above construction is 60 degree.
  6. We get ∠BOA which measures 60 degrees.

2. Construction of 30° Angle

  1. Draw a line with the ruler. Keep the end of the compass at one end of the line and the pencil on the other end of the line.
  2. With the same measure draw, two arcs from both the ends of the line. You will get an equilateral triangle.
  3. Now join the line AC. Here we get a 60° angle.
  4. Now, draw an angle bisector. From the point, A draw an arc on AB and AC and from these arcs draw another arc at the centre as shown in the above figure.
  5. Here we get a 30°
  6. Each angle is 30°

3. Construction of 120° Angle

  1. Take a ruler and draw a line OA of any convenient length.
  2. Now take a compass and take any convenient radius and taking O as the centre draw an arc which cuts the line segment OA at B as shown in the above figure.
  3. Take the compass again and with the same radius, take B as the centre draw an arc which cuts arc at C
  4. Now with the same radius take a centre C to draw another arc which cuts the first arc at D
  5.  Now join OD and extend it to E
  6. So now we get the ∠EOA = 120°

Solved Example For You

Question 1. Which of the following angle is possible to construct using a compass?

  1. 60 °
  2. 32 °
  3. 51.25 °
  4. 40 °

Answer : A. 60 ° angle is possible to construct using a compass.

Question 2: What is meant by angle of bisector?

Answer: Angle bisector of a triangle refers to a line segment that carries out bisection of one of the vertex angles of a triangle.

Question 3: Can we say that a bisector cuts an angle in half?

Answer: Yes, angle bisector is the line that cuts the angle in half.

Question 4: Explain the way for bisecting a 45-degree angle?

Answer: The way for bisecting a 45-degree angle consists of the following steps:

  • Draw a line segment of a suitable length.
  • Now, draw a 90° with compass.
  • With the dotted lines, draw the perpendicular.
  • Draw 45° by compass.
  • From the point where the 90° was drawn, draw two arcs of the same radius.
  • Now draw two more arcs by making use of those arcs such that their intersection takes place at some point.
  • Finally, join the central point with the intersection point.

Question 5: Differentiate between a perpendicular bisector and an angle bisector?

Answer: An angle bisector causes division of an angle into two congruent angles while a perpendicular bisector causes splitting of a segment into two congruent segments.

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