Mathematics

In the figure, three coplanar lines $$AB,CD$$ and $$EF$$ intersect at a point $$O$$, forming angles as shown. Find the values of $$x,y,z$$ and $$t$$


SOLUTION
From the figure we know that $$\angle COE$$ and $$\angle DOF$$ are vertically opposite angles

$$\angle COE=\angle DOF=\angle z={50}^{o}$$

From the figure we know that $$\angle BOD$$ and $$\angle COA$$ are vertically opposite angles

$$\angle BOD=\angle COA=\angle t={90}^{o}$$

We also know that $$\angle COA$$ and $$\angle AOD$$ form a linear pair

$$\angle COA+\angle AOD={ 180 }^{ o }$$

It can also be written as

$$\angle COA+\angle AOF+\angle FOD={ 180 }^{ o }$$

By substituting the values we get

$${90}^{o}+{x}^{o}+{50}^{o}={ 180 }^{ o }$$

$${x}^{o}={40}^{o}$$

From the figure we know that $$\angle EOB$$ and $$\angle AOF$$ are vertically opposite angles

$$\angle EOB=\angle AOF=x=y=40$$

Therefore the values of $$x,y,z,t$$ are $$40^{o},40^{o},50^{o},90^{o}$$
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