1. Construct an angle of 45° at the initial point of a given ray and justify the construction
Answers
Question:-
Construct an angle of 45° at the initial point of a given ray and justify the construction.:-
Answer:-
The below given steps will be followed to construct an angle of 45°.
(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
(iv) Taking S and T as centre, draw an arc of same radius to intersect each other at U.
(vi) From R and V, draw arcs with radius more than RV/2 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.
So, ∠UPS = ∠TPS/2 = 60°/2 = 30°
Also, ∠UPQ = ∠SPQ + ∠UPS
In step (vi) of this construction, PW was constructed as the bisector of ∠UPQ.
So, ∠WPQ = ∠UPQ/2 = 90°/2 = 45°