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Draw $ \angle \mathrm{POQ} $ of measure $ 75^{\circ} $ and find its line of symmetry.
To do:
We have to draw $\angle{PQR}$ of measure $75^o$ and find its line of symmetry.
Solution:
Steps of construction:
(i) Let us draw a line $l$ of any length and mark two points $O$ and $Q$ at any distance from each other on the line $l$.
(ii) Now, by taking compasses draw an arc of any length $l$ from point $O$ and mark the point of intersection of the arc with line $l$ as $R$.
(iii) Now, by taking compasses let us draw an arc with the same radius as before from point $R$ and
let us mark the point of intersection of this arc with the previous arc as point $S$.
(iv) Now, by taking the compasses let us draw another with the same radius as before from point $S$ and mark the point of intersection of this arc with the previous arc as $T$
(v) Now, by taking the compasses with the same radius as before let us draw an arc from points $P$ and $S$ and let us mark the point of intersection of these arcs as $U$ and let us join the points $O$ and $U$.
(vi) $\overline{OU}$ intersects arc at a point let us name this point of intersection as $V$.Now, by taking compasses with a radius of more than half of the radius from point $S$ to $V$ let us draw another arc from points $S$ and $V$ these arcs intersect each other at a point let us name it $P$.
(vii) Now, let us join point $P$ with point $O$. Thus, $\overline{OP}$ is the required line of $75^o$.
(viii) $\overline{OP}$ intersects the arc drawn from $O$ at a point let us mark this point of intersection as $W$.
(ix) Now, by taking compasses let us draw another arc with a radius greater than half of the length of $RW$ from points $R$ and $W$ within the $75^o$ angle to the line $l$ Now let us mark the point intersection of this arc as$X$ and lets us join point $O$ and $X$
(x) Therefore, $\overline{OX}$ is the symmetry of $\angle{PQR}=75^o$.
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