Mathematics

In the figure below, $$AA'$$ is parallel to $$CC'$$. The measure $$w$$ of $$\angle A'AB$$ is equal to $$135$$ degrees and the measure $$z$$ of $$\angle C'CB$$ is equal to $$147$$ degrees. Find $$\angle ABC$$. 


ANSWER

$$78^o$$


SOLUTION
Draw $$BB'$$ parallel to $$AA'$$ and $$CC'$$as shown in the figure below.
Note that angle $$ABC$$ is given by 
$$\angle ABC = \angle ABB' + \angle CBB' $$
$$\angle w'$$ and $$\angle ABB'$$ are alternate interior angles and their measure are equal. 
$$\angle ABB' = \angle w' $$
$$\angle z'$$ and $$\angle CBB'$$ are alternate interior angles and their sizes are equal. 
$$\angle CBB' = \angle z'$$
Angles $$w$$ and $$w'$$ are supplementary which gives 
$$w' = 180^o - w = 180^o - 135^o = 45^0$$
Angles $$z$$ and $$z'$$ are also supplementary which gives 
$$z' = 180^o - z = 180^o - 147^o = 33^o$$
We now substitute $$\angle ABB'$$ by $$w'$$ and $$\angle CBB'$$ by $$z'$$ in $$\angle ABC = \angle ABB' + \angle CBB'$$ found above. 
$$\angle ABC = w' + z' = 45^o + 33^o = 78^o$$
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