Construction of lines and angles

# Construction of lines and angles

## Construction of lines and angles in mathematics is an important topic. In this article, you will be able to learn the constructions of different types of lines and angles. All types of angles can be constructed using a protractor. Only a few standard angles such as 30°, 45°, 60°, 90°, etc. can be constructed using a pair of compasses.

Let us learn how to construct parallel lines and perpendicular lines. After that, we will learn constructing standard angles using a pair of compasses.

## Construction of a Perpendicular to a Given Line Segment

Steps of Construction

1.      Draw a line segment AB and take any point C on it.

2.      Put a pair of compasses at point C and draw arcs cutting AB at P and Q.

3.      Again open the pair of compasses at a convenient radius and draw two arcs
with centers P and Q and intersecting each other at R.

4.      Join CR.

CR is the required perpendicular to AB at C.

## Construction of a Perpendicular to a Given Line Segment from a Point Outside the Line

Steps of Construction

1.      Draw a line segment AB and take a point O anywhere outside the line segment AB.

2.      Put a pair of compasses at point O. Open the pair of compasses and draw an arc in such a way that AB is intersected at two points C and D.

3.      Again, with C and D as centers, draw two arcs (keeping the same radius) intersecting each other at E. Join O to E.

OE is the required perpendicular to AB.

## Construction of Perpendicular Bisector to a Given Line Segment

Steps of Construction

1.      Draw a line segment AB on a sheet of paper.

2.      Open a pair of compasses to a little more than half of the length of AB and with A as the center, draw two arcs above and below AB.

3.      Keeping the same radius, place the pair of compasses at B and draw two arcs above and below AB. Let the arcs cut the previous arcs at P and Q.

4.      Join P and Q intersecting AB at O.

PQ is the required perpendicular bisector of the line segment AB.

## Construction of a Line Parallel to a Given Line at a Given Distance

Example: Draw a line parallel to a given line at a distance of 4.5 cm.

Steps of Construction

1.      Draw a line l on a sheet of paper. Take any point L on line l.

2.      Draw a line m perpendicular to l from point L. Cut off LM = 4.5 cm.

3.      At the point M, draw MP perpendicular to line m.

MP is the required line parallel to l at a distance of 4.5 cm.

## Construction of an Angle Equal to a Given Angle

Example: Construct TLM equal to ABC.

Steps of Construction

1.      Draw a ray LM.

2.      With B as the center, draw an arc with any convenient radius touching BC at P and BA at Q respectively.

3.      With L as the center and keeping the same radius, draw an arc intersecting the ray LM at N.

4.      With P as the center, draw an arc which meets BA at Q.

5.      With N as the center and radius equal to PQ, draw an arc intersecting the arc in step 3 at S. Join LS and produce it to T.

TLM is the required angle equal to ABC.

## Construction of a Bisector of a Given Angle

Example: Construct the bisector of ABC.

Steps of Construction

1.      Draw an ABC of any measurement.

2.      With B as the center and any convenient radius draw an arc intersecting the two arms of the angle at D and E respectively.

3.      With centers D and E and radius equal to more than DE, draw two arcs intersecting each other at point O. Join BO and produce it to F.

BF is the bisector of ABC.

## Construction of an Angle of 60°

Steps of Construction

1.      Draw a ray OA.

2.      With O as the center and any convenient radius, draw an arc cutting OA at C.

3.      With C as the center and keeping the same radius, draw an arc cutting the previous arc at D. Join OD and produce it to B.

AOB is the required angle of 60°.

## Construction of an Angle of 120°

Steps of Construction

1.      Draw a ray OB.

2.      With O as the center and any convenient radius, draw an arc cutting the ray OB at P.

3.      With P as the center and keeping the same radius, draw an arc to cut the above arc at Q.

4.      With Q as the center and keeping the same radius, draw an arc to cut the arc at R.

5.      Join OR and produce it to A.

AOB is the required angle of 120°.

## Construction of an Angle of 90°

Steps of Construction

1.      Draw a ray OA.

2.      Put the pair of compasses at point O. Take any suitable radius and draw an arc which cuts OA at point C.

3.      With point C as the center and the same radius, mark an arc and name it as D. Again put the pair of compasses at D and with the same radius, draw an arc which intersects the initial arc at E.

4.      Put the pair of compasses at E and draw an arc with a convenient radius. Repeat the process by putting the pair of compasses at D and let the two arcs intersect at B. Join OB.

BOA is the required angle of 90°.

## Construction of an Angle of 45°

Steps of Construction

1.      Draw an angle AOB = 90° as described above.

2.      Bisect AOB using the steps of constructing an angle bisector.

Hence, AOF is the required angle of 45°.

## Construction of an Angle of 135°

Steps of Construction

1.      Draw a straight line l and take a point O on it.

2.      Construct BOA = 90° as explained above. Bisect GOB.

AOH is the required angle of 135°.

## Construction of a Tangent

Example: Construct a tangent to a circle of radius 2.5 cm at a point A.

Steps of Construction

1.      By taking O as the center and radius equal to 2.5 cm, draw a circle.

2.      Mark a point A on the circle and join OA to get the radius.

3.      Draw a perpendicular line l at point A and mark a point B on it.

AB is the required tangent.

## Circumcircle of a Triangle

Example: Construct a circumcircle of an equilateral triangle PQR.

Steps of Construction

1.      Draw an equilateral triangle PQR.

2.      Draw perpendicular bisectors AB and CD of sides PQ and QR respectively.

3.      Let AB and CD intersect at O.

4.      Taking O as center and radius OQ, draw a circle which passes through P, Q and R.

This is the required circumcircle.

## Incircle of a Triangle

Example: Construct an incircle of an isosceles triangle ABC.

Steps of Construction

1.      Draw an isosceles triangle ABC.

2.      Draw the angle bisectors of A and B and name them AY and BQ respectively.

3.      Let AY and BQ intersect at O.

4.      Draw a perpendicular ON on BC and with O as center and radius equal to ON, draw a circle which touches all the sides of the triangle ABC.

This is the required incircle.