Question
Construct with ruler and compasses, angles of measure: $45^\circ $
✓
Answer
The below given steps will be followed to construct an angle of $45^\circ .$
$i.\ $Draw a line $l$ and mark a point $P$ on it. Now taking $P$ as centre and with a convenient radius, draw an arc of a circle which intersects line $l$ at $Q.$
$ii.\ $Taking $Q$ as centre and with the same radius as before, draw an arc intersecting the previously drawn arc at $R.$
$iii.\ $Taking $R$ as centre and with the same radius as before, draw an arc intersecting the arc at $S ($see figure$).$
$iv.\ $Taking $R$ and $S$ as centres, draw arcs of same radius to intersect each other at $T.$
$v.\ $Join $PT.$ Let it intersect the major arc at point $U.$
$vi.\ $Taking $Q$ and $U$ as centres, draw arcs with radius more than $\frac{1}{2}QU$ to intersect each other at $V.$ Join $PV.$
$PV$ is the required ray making $45^\circ $ with the given line $l.$
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