Construction of Angles

Summary

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Construction of copy of an angle:

An exact copy of a line segment can be constructed using a ruler and a compass.

  • Construction of a copy of an angle YXZ.
  • Draw a line AB.
  • Mark any point O on AB.
  • Place the compass pointer at vertex X of the given figure and draw an arc with a convenient radius, cutting rays XY and XZ at points E and F, respectively.
  • Without changing the compass settings, draw an arc on line AB from point O. It cuts line AB at P.
  • Set the compass to length EF.
  • Without changing the compass settings, draw an arc from P cutting the previous arc at point Q.
  • Join points O and Q.
  • Hence, ∠POQ  is the required copy of ∠YXZ.



Construction of the bisector of an angle:
  • Construction of the bisector of an angle LMN.
  • Place the compass pointer at vertex M of the given angle.
  • Draw an arc cutting rays ML and MN at U and V respectively.
  • Draw an arc with V as the centre and a radius more than half the length of UV in the interior of ∠LMN.
  • Draw another arc with U as the centre and the same radius intersecting the previous arc.
  • Name the point of intersection of the arcs as X.
  • Join points M and X.
  • Ray MX is the required bisector of ∠LMN.


Construction of a 60° angle:

  • Draw a line.
  • Mark a point P on the line.
  • Draw an arc from point P with a convenient radius cutting the line at a point.
  • Name the point of intersection of the arc and the line as Q.
  • Draw another arc with Q as the centre and the same radius so that it passes through point P.
  • Name the point of intersection of the two arcs as R.
  • Join points P and R.
  • Ray PR forms an angle with ray PQ at point P, which measures 60°.
  • ∠QPR is the required angle measuring 60°.


Construction of a 30° angle:

  • To obtain a 30° angle, we need to bisect a 60° angle.
  • Draw an arc with Q as the centre and a radius more than half the length of QR.
  • Draw another arc with R as the centre without changing the compass settings so that it intersects the previous arc.
  • Name the point of intersection of the arcs as S.
  • Join points P and S.
  • ∠QPS is the required angle measuring 30°.

In a similar way, we can construct a 120° angle and 90° angle without using the protractor.


Construction of a 120° angle
  • Draw line XY.
  • Mark a point on the line and name it as P.
  • Draw an arc with P as the centre and a convenient radius so that it cuts the line at Q.
  • Draw another arc with Q as the centre without changing the compass settings so that it intersects the first arc at R.
  • Draw another arc with R as the centre without changing the compass settings so that it intersects the first drawn arc at point S.
  • Join points P and S.
  • ∠SPQ is the required angle measuring 120°.


Construction of a 90° angle

  • Draw line l and mark point P on it.
  • Draw an arc with P as the centre and a convenient radius cutting line l at Q.
  • Draw another arc with Q as the centre and the same radius cutting the first arc at R.
  • Draw an arc with R as the centre and the same radius cutting the first arc at S.
  • Join points P and R.
  • Join points P and S.
  • 90° lies to the centre of 60° and 120°.
  • Draw an arc with R as the centre and a radius more than half the length of RS in the interior of ∠RPS.
  • Draw another arc with S as the centre and the same radius so that it intersects the previous arc at T.
  • Join points P and T.
  • PT is the perpendicular line to PQ.
  • ∠QPT is the required angle measuring 90°.

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