Question
In the given figure, if $\text{AB }||\text{ DE}$ and $\text{BD }||\text{ FG}$ such that $\angle\text{FGH}=125^\circ$ and $\angle\text{B}=55^\circ,$ find x and y.
✓
Answer
In the given figure, if$\text{AB }||\text{ DE},\text{BD }||\text{ FG},\angle\text{FGH}=125^\circ$ and$\angle\text{B}=55^\circ$ We need to find the value of x and y 
Here, as AB || DE and BD is the transversal, so according to the property, "alternate interior angles are equal", we get$\angle\text{D}=\angle\text{B}$
$\angle\text{D}=55^\circ\dots(1)$
Similarly, as BD || FG and DF is the transversal$\angle\text{D}=\angle\text{F}$
$\angle\text{F}=55^\circ(\text{Using 1})$
Further, EGH is a straight line. So, using the property, angles forming a linear pair are supplementary$\angle\text{FGE}=\angle\text{FGH}=180^\circ$
$\text{y}+125^\circ=180^\circ$
$\text{y}=180^\circ-125^\circ$
$\text{y}=55^\circ$
Also, using the property, "an exterior angle of a triangle is equal to the sum of the two opposite interior angles", we get, In $\triangle\text{EFG}$ with $\angle\text{FGH}$ as its exterior angle$\text{ext.}\angle\text{FGH}=\angle\text{F}+\angle\text{E}$
$125^\circ=55^\circ+\text{x}$
$\text{x}=125^\circ-55^\circ$
$\text{x}=70^\circ$
Thus, $\text{x}=70^\circ$ and $\text{y}=55^\circ$
Here, as AB || DE and BD is the transversal, so according to the property, "alternate interior angles are equal", we get$\angle\text{D}=\angle\text{B}$$\angle\text{D}=55^\circ\dots(1)$
Similarly, as BD || FG and DF is the transversal$\angle\text{D}=\angle\text{F}$
$\angle\text{F}=55^\circ(\text{Using 1})$
Further, EGH is a straight line. So, using the property, angles forming a linear pair are supplementary$\angle\text{FGE}=\angle\text{FGH}=180^\circ$
$\text{y}+125^\circ=180^\circ$
$\text{y}=180^\circ-125^\circ$
$\text{y}=55^\circ$
Also, using the property, "an exterior angle of a triangle is equal to the sum of the two opposite interior angles", we get, In $\triangle\text{EFG}$ with $\angle\text{FGH}$ as its exterior angle$\text{ext.}\angle\text{FGH}=\angle\text{F}+\angle\text{E}$
$125^\circ=55^\circ+\text{x}$
$\text{x}=125^\circ-55^\circ$
$\text{x}=70^\circ$
Thus, $\text{x}=70^\circ$ and $\text{y}=55^\circ$
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