Mathematics

Two lines $$AB$$ and $$CD$$ intersect each other at a point $$O$$ such that $$\angle AOC:\angle AOD=5:7$$. Find all the angles.


SOLUTION
Consider $$\angle AOC$$ as $$5x$$ and $$\angle AOD$$ as $$7x$$
from the figure we know that $$\angle AOC$$ and $$\angle AOD$$ for linear pair of angle

so it can be written as

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

by substituting the values

$$5x+7x={ 180 }^{ o }$$

$$x=180/12$$

$$x={15}^{o}$$

by susbtituting the value of $$x$$

$$\angle AOC=5x$$

so we get

$$\angle AOC=5(15)={75}^{o}$$

$$\angle AOD=7x$$

so we get

$$\angle AOD=7(15)={105}^{o}$$

from the figure we know that $$\angle AOC$$ and $$\angle BOD$$ are vertical angles 

so we get

$$\angle AOC=\angle BOD={75}^{o}$$

from the figures we know that $$\angle AOD$$ and $$\angle BOC$$ are vertical angles 

so we get

$$\angle AOD=\angle BOC={105}^{o}$$
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