**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.

**Step 2:**Taking D as the centre and any radius more than half of DE, draw an arc.

**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.

Thus, ray AQ is the required bisector of ∠CAB or ∠BAC.

**Read More:**

- Construction of an Equilateral Triangle
- Construction Of Similar Triangle As Per Given Scale Factor
- Construction Of A Line Segment
- Construction Of Perpendicular Bisector Of A Line Segment
- Construction Of An Angle Using Compass And Ruler

**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°.

**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:

**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.

**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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