Mathematics

In the given figure line AB and CD intersect at O . $$\angle BOC = {36^ \circ }$$. Find $$\angle x$$ , $$\angle y$$ and  $$\angle z$$ 


SOLUTION
In first figure,
$$\angle AOD=\angle BOC$$ ------- Vertically Opposite angles
$$\therefore x= 36^\circ$$

$$\angle AOC+\angle COB=180^\circ$$ ------ Angles on straight line
$$y+36^\circ=180^\circ$$
$$y=180^\circ-36^\circ$$
$$y=144^\circ$$
$$\angle AOC=\angle BOD$$ ------- Vertically Opposite angles
$$y=z=144^\circ$$


In second figure,
$$\angle POR+\angle ROT+\angle QOT=180^\circ$$ ------ Angles on straight line
$$2x+90^\circ+x=180^\circ$$
$$3x=90^\circ$$
$$x=30^\circ$$
$$\therefore 2x= 60^\circ$$

$$\angle POR=\angle SOQ$$ ------- Vertically Opposite angles
$$2x=y=60^\circ$$

$$\angle POS=\angle ROQ$$ ------- Vertically Opposite angles
$$z=90^\circ+x$$
$$z=90^\circ+30^\circ$$
$$z=120^\circ$$
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