Construct an angle of 45° at the
initial point of a given ray and justify the construction.
The below given steps will be followed to construct an angle of
(i) Take the given ray PQ. Draw an arc of some radius taking
point P as its centre, which intersects PQ at R.
(ii) Taking R as centre and with the same radius as before, draw
an arc intersecting the previously drawn arc at S.
(iii) Taking S as centre and with the same radius as before, draw
an arc intersecting the arc at T (see figure).
(iv) Taking S and T as centre, draw an arc of same radius to
intersect each other at U.
(v) Join PU. Let it intersect the arc at point V.
(vi) From R and V, draw arcs with radius more than
RV to intersect each other at W. Join PW.
PW is the required ray making 45° with
Justification of Construction:
We can justify the construction, if we can prove
∠WPQ = 45°.
For this, join PS and PT.
We have, ∠SPQ =
∠TPS = 60°.
In (iii) and (iv) steps of this construction, PU was drawn as the
bisector of ∠TPS.
Also, ∠UPQ =
= 60° + 30°
In step (vi) of this construction, PW was constructed as the
bisector of ∠UPQ.
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