How Do You Construct A Bisector Of An Angle

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Construction Of The Bisector Of A Given Angle

Given: An angle CAB
To construct: Bisector of ∠CAB.

  • Step 1: Taking A as the centre and with any suitable radius, draw an arc cutting the arms AB and AC of ∠CAB at D and E respectively.
    Construction-of-Bisector-of-Given-Angle
  • Step 2: Taking D as the centre and any radius more than half of DE, draw an arc.
    Construction-of-Bisector-of-Given-Angle-2
  • Step 3: Similarly, taking E as the centre and with the same radius (as in step 2), draw an arc intersecting the previous arc at P. Join AP and produce it to get AQ.
    Construction-of-Bisector-of-Given-Angle-3 Thus, ray AQ is the required bisector of ∠CAB or ∠BAC.

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Example 1:    Using a protractor, draw an angle of measure 78°. With this angle as given, draw an angle of measure 39°.
Solution:    We follow the following steps to draw an angle of 39° from an angle of 78°.
Construction-Of-Bisector-Of-An-Angle-Example-1 Steps of Construction:

  • Step I: Draw a ray OA as shown in fig.
  • Step II: With the help of a protractor construct an angle AOB of measure 78°.
  • Step III: With centre O and a convenient radius drawn an arc cutting sides OA and OB at P and Q respectively.
  • Step IV: With centre P and radius more than 1/2 (PQ), drawn an arc.
  • Step V: With centre Q and the same radius, as in the previous step, draw another arc intersecting the arc drawn in the previous step at R.
  • Step VI: Join OR and produce it to form ray OX.
    The angle ∠AOX so obtained is the required angle of measure 39°.

Verification: Measure ∠AOX and ∠BOX. You will find that
∠AOX = ∠BOX = 39°.

Example 2:    Using a protractor, draw an angle of measure 128º. With this angle as given, draw an angle of measure 96º.
Solution:    In order to construct an angle of measure 96º from an angle of measure 128º, we follow the following steps:
Construction-Of-Bisector-Of-An-Angle-Example-2 Steps of Construction: 

  • Step I: Draw an angle ∠AOB of measure 128º by using a protractor.
  • Step II: With centre O and a convenient radius draw an arc cutting OA and OB at P and Q respectively.
  • Steps III: With centre P and radius more than 1/2 (PQ), draw an arc.
  • Step IV: With centre Q and the same radius, as in step III, draw another arc intersecting the previously drawn arc at R.
  • Steps V: Join OR and produce it to form ray OX. The ∠AOX so obtained is of measure (128º/2) i.e. 64º.
  • Step VI: With centre S (the point where ray OX cuts the arc (PQ) and radius more than 1/2 (QS), draw an arc.
    Construction-Of-Bisector-Of-An-Angle-Example-2-1
  • Step VII: With centre Q and the same radius, as in step VI, draw another arc intersecting the arc drawn in step VI at T.
  • Step VIII: Join OT and produce it form OY.
    Clearly, ∠XOY = 1/2 ∠XOB = 1/2 (64º) = 32º.
    ∴ ∠AOT = ∠AOX + ∠XOY = 64º + 32º = 96º
    Then, ∠AOY is the desired angle.

Verification: Measure ∠AOX, ∠XOY and ∠AOY. You will find ∠AOY = 96º.

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