Summary

Videos

References

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, âˆ POQ is the required copy of âˆ YXZ.

- 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 âˆ 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 âˆ LMN.

**Construction of a 60Â° 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 60Â°.
- âˆ QPR is the required angle measuring 60Â°.

**Construction of a 30Â° angle:**

- To obtain a 30Â° angle, we need to bisect a 60Â° 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.
- âˆ QPS is the required angle measuring 30Â°.

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

- 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.
- âˆ SPQ is the required angle measuring 120Â°.

**Construction of a 90Â° 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.
- 90Â° lies to the centre of 60Â° and 120Â°.
- Draw an arc with R as the centre and a radius more than half the length of RS in the interior of âˆ 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.
- âˆ QPT is the required angle measuring 90Â°.

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, âˆ POQ is the required copy of âˆ YXZ.

- 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 âˆ 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 âˆ LMN.

**Construction of a 60Â° 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 60Â°.
- âˆ QPR is the required angle measuring 60Â°.

**Construction of a 30Â° angle:**

- To obtain a 30Â° angle, we need to bisect a 60Â° 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.
- âˆ QPS is the required angle measuring 30Â°.

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

- 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.
- âˆ SPQ is the required angle measuring 120Â°.

**Construction of a 90Â° 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.
- 90Â° lies to the centre of 60Â° and 120Â°.
- Draw an arc with R as the centre and a radius more than half the length of RS in the interior of âˆ 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.
- âˆ QPT is the required angle measuring 90Â°.