Mathematics

# In the figure given below, $AB, EF$ and $CD$ are parallel lines. Given that $AB =15\ cm, EG = 5\ cm, GC = 10\ cm$ and $DC = 18 cm$. Calculate(i) $EF$(ii) $AC$.

##### SOLUTION
Consider $\triangle EFG$ and $\triangle GCD$
$EF\parallel CD\implies \angle GCD=\angle GEF\ (\because$ alternative angles $)$
$\angle FGE=\angle CGD\ (\because$ vertically opposite angles $)$
as two angles are equal automatically third angle will be equal
So by AAA criteria , we get
$\triangle EFG\sim \triangle CDG$
$\implies \dfrac{EF}{EG}=\dfrac{CG}{CD}\implies EF=5\times \dfrac{18}{10}=9\ cm$
$AB\parallel EF\implies \angle BAC=\angle FEC,\angle ABC=\angle EFC$ and $\angle C$ remains the same
So by AAA criteria ,we get
$\triangle ABC\sim \triangle EFC$
$\implies \dfrac{AC}{AB}=\dfrac{EC}{EF}\implies AC=15\times \dfrac{15}{9}=25\ cm$

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