Construction of Angles

Objective

This topic gives an overview of; 

  • Angles
  • Constructing an Angle of a given Measure
  • Constructing a copy of an angle of unknown measure
  • Bisector of an angle 
  • Identifying an angle
  • Measure of an angle

Angles

An angle is a shape, formed by two lines or rays diverging from a common point (the vertex).

Constructing an Angle of a given Measure

Suppose we want an angle of measure 40°.

 Here are the steps to follow:

Step 1: Draw AB of any length. 

Step 2: Place the centre of the protractor at A and the zero edge along AB.

 

Step 3: Start with zero near B. Mark point C at 40°.

 

Step 4: Join AC. ∠BAC is the required angle.

Constructing a copy of an angle of unknown measure

Suppose an angle (whose measure we do not know) is given and we want to make a copy of this angle. As usual, we will have to use only a straight edge and the compasses. Given ∠A, whose measure is not known.

Step 1: Draw a line l and choose a point P on it.

Step 2: Place the compasses at A and draw an arc to cut the rays of ∠A at B and C.

Step 3: Use the same compasses setting to draw an arc with P as centre, cutting l in Q.

Step 4: Set your compasses to the length BC with the same radius.

Step 5: Place the compasses pointer at Q and draw the arc to cut the arc drawn earlier in R.

Step 6: Join PR. This gives us ∠P. It has the same measure as ∠A.

This means ∠QPR has same measure as ∠BAC

 Bisector of an angle

Mark a point O on it. With O as initial point, draw two rays OA and OB. You get ∠AOB. Fold the sheet through O such that the rays OA and OB coincide. Let OC be the crease of paper which is obtained after unfolding the paper. OC is clearly a line of symmetry for ∠AOB. Measure ∠AOC and ∠COB. OC the line of symmetry is therefore known as the  angle bisector of ∠AOB.

Construction with Ruler and Compasses

Let an angle, say, ∠A be given.

Step 1: With A as centre and using compasses, draw an arc that cuts both rays of ∠A. Label the points of intersection as B and C.

Step 2: With B as centre, draw (in the interior of ∠A) an arc whose radius is more than half the length BC.

Step 3: With the same radius and with C as centre, draw another arc in the interior of ∠A. Let the two arcs intersect at D. Then AD is the required bisector of ∠A.

Identifying an angle

Vertex: The vertex is the common point at which the two lines or rays are joined. Point B is the figure above is the vertex of the angle ∠ABC.

Legs: The legs (sides) of an angle are the two lines that make it up. In the figure above, the lines AB and BC are the legs of the angle ∠ABC.

Interior: The interior of an angle is the space in the 'jaws' of the angle extending out to infinity. 

Exterior: All the space on the plane that is not the interior. 

An angle can be identified in two ways.

  1. Like this: ∠ABC 

    The angle symbol, followed by three points that define the angle, with the middle letter being the vertex, and the other two on the legs. So in the figure above the angle would be ∠ABC or ∠CBA. So long as the vertex is the middle letter, the order is not important. As shorthand we can use the 'angle' symbol. For example '∠ABC' would be read as 'the angle ABC'.
  2. Or like this: ∠B

    Just by the vertex, so long as it is not ambiguous. So in the figure above the angle could also be called simply '∠B'

Measure of an angle

The size of an angle is measured in degrees. When we say 'the angle ABC' we mean the actual angle object. If we want to talk about the size, or measure, of the angle in degrees, we should say 'the measure of the angle ABC' - often written m ∠ABC.

However, many times we will see '∠ABC=34°'. Strictly speaking this is an error. It should say 'm ∠ABC=34°' 

Angles of special measures

There are some elegant and accurate methods to construct some angles of special sizes which do not require the use of the protractor. We discuss a few here.

Constructing a 60° angle

Step 1:Draw a line l and mark a point O on it.

Step 2: Place the pointer of the compasses at O and draw an arc of convenient radius which cuts the line PQ at a point say, A.

Step 3: With the pointer at A (as centre),now draw an arc that passes through O.

Step 4: Let the two arcs intersect at B. Join OB. We get ∠BOA whose measure is 60°.

Constructing a 30° angle

Construct an angle of 60° as shown earlier. Now, bisect this angle. Each angle is 30°, verify by using a protractor.

Constructing a 120° angle

An angle of 120° is nothing but twice of an angle of 60°.Therefore, it can be constructed as follows :

Step 1: Draw any line PQ and take a point O on it.

Step 2: Place the pointer of the compasses at O and draw an arc of convenient radius which cuts the line at A.

Step 3: Without disturbing the radius on the compasses, draw an arc with A as centre which cuts the first arc at B.

Step 4: Again without disturbing the radius on the compasses and with B as centre, draw an arc which cuts the first arc at C.

Step 5: Join OC, ∠COA is the required angle whose measure is 120°.

Constructing a 90° angle

Construct a perpendicular to a line from a point lying on it, as discussed earlier. This is the required 90° angle.

Summary

  • Using the ruler and compasses, the following constructions can be made:
    • An angle of a given measure.
    • A copy of an angle.
    • The bisector of a given angle. Some angles of special measures such as  90°, 45°, 60°,30°,120°,35°.
  • The angle symbol, followed by three points that define the angle, with the middle letter being the vertex, and the other two on the legs.
  • The size of an angle is measured in degrees.

 

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