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 $\angle ABC$.

$78^o$

##### 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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Single Correct Medium Published on 09th 09, 2020
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