In the figure,$AB$,$CD$,and $EF$ are three lines concurrent at $O$. Find the value of $y$. (in $^{\circ}$)

$\angle ABD$ and $\angle ACE$ are exterior angles of $\Delta ABC$. If $\angle ABD = 140^{\circ}$ and $\angle ACE = 80^{\circ}$,then find $\angle A$. (in $^{\circ}$)

In $\Delta ABC$,the sides $AB$ and $AC$ are produced to $D$ and $E$ respectively,so that exterior angles $\angle CBD$ and $\angle BCE$ are formed. If bisectors of $\angle CBD$ and $\angle BCE$ intersect at point $O$,prove that $\angle BOC = 90^{\circ} - \frac{1}{2} \angle A$.

In the figure,$OD$ is the bisector of $\angle AOC$,$OE$ is the bisector of $\angle BOC$,and $OD \perp OE$. Show that the points $A, O$,and $B$ are collinear.

If one angle of a triangle is equal to the sum of the other two angles,then the triangle is

In the given figure,$POQ$ is a straight line. The value of $x$ is: (in $^{\circ}$)