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Asked by Kalpana Last Modified
Answer 
UPQ = 90o.
SPQ = TPS = 60o. In (iii) and (iv) steps of this construction, we have drawn PU as the bisector of TPS.
Tasneem
The steps given to construct an angle of 45° is as follows.
(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 PQ.
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.
∴ ∠UPS = 
∠TPS
Also, ∠UPQ = ∠SPQ + ∠UPS
= 60° + 30°
= 90°
In step (vi) of this construction, PW was constructed as the bisector of ∠UPQ.
∴ ∠WPQ = ∠UPQ 
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