$O$ is the incenter of the $∆ABC$ and $\angle A={30}^{\mathrm{o}}$, then $\angle BOC$ is:

In the adjoining figure $AB\parallel CD$ and $PQ,$ $QR$ intersect $AB$ and $CD$ both at $E, F, G$ and $H$ respectively. Given that $m\angle PEB={80}^{\mathrm{o}}$, $m\angle QHD={120}^{\mathrm{o}}$ and $\angle PQR=x^{\mathrm{o}}$, find the value of $x.$

Two parallel lines $AB$ and $CD$ are intersected by a transversal $EF$ at $M$ and $N$ respectively. The lines $MP$ and $NP$ are the bisectors of interior angle $\angle BMN$ and $\angle DNM$ on the same side of the transversal. Then, $\angle MPN$ is equal to

In the given figure, $△ABC$ is an isosceles triangle in which $AB=AC$ and $\angle ABC={50}^{\mathrm{o}}$, then $\angle BDC$ is equal to:

In the given figure, $PT$ is the tangent of a circle with centre $O$ at point $R$. If diameter $SQ$ is increased, it meets with $PT$ at point $P.$ If $\angle SPR=x^{\mathrm{o}}$ and $\angle QSR=y^o$, what is the value of $x^{\mathrm{o}}+2y^{\mathrm{o}}$?

In the diagram $AB\parallel GH\parallel DE$ and $GF\parallel BD\parallel HI,$$\angle \mathrm{FGC}={80}^{\mathrm{o}}$, find the value of $\angle CHI.$

In the given figure, $\angle PQR={90}^{\mathrm{o}}$. $O$ is the centroid of the $∆PQR$, where $PQ=5$ cm and $QR=12$ cm, then $OQ$ is equal to: