Mathematics

In the figure, three coplanar lines $$AB,CD$$ and $$EF$$ intersect at a point $$O$$. Find the value of $$x$$. Hence, find $$\angle AOD,\angle COE$$ and $$\angle AOE$$


SOLUTION
From the figure we know that $$\angle COE$$ and $$\angle EOD$$ form a linear pair

$$\angle COE+\angle EOD={ 180 }^{ o }$$

It can also be written as

$$\angle COE+\angle EOA+\angle AOD={ 180 }^{ o }$$

By substituting values in the equation we get

$$5x+\angle EOA+2x={ 180 }^{ o }$$

From the figure we know that $$\angle EOA$$ and $$\angle BOF$$ are vertically opposite angles

$$\angle EOA=\angle BOF$$

so we get

$$5x+\angle BOF+2x={180}^{o}$$

$$5x+3x+2x={ 180 }^{ o }$$

$$10x={ 180 }^{ o }$$

$$x=18$$

By substituting the value of $$x$$

$$\angle AOD=2{x}^{o}$$

$$\angle AOD=2(18)={36}^{o}$$

$$\angle EOA=\angle BOF=3{x}^{o}$$

So we get

$$\angle EOA=\angle BOF=3(!8)={54}^{o}$$

$$\angle COE=5{x}^{o}$$

so we get

$$\angle COE=5(18)={90}^{o}$$
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