Question

In the given figure, $AB||FH$, $DF||BG$, the which of the following is correct?

• A. $△ADE~△HGC$
• B. $△ABC~△HFE$
• C. $AE.FE=DE.EH$
• D. $AB.HF=BC.FE$

Answer 1

Answer: (A), (B), (C)

$AE.FE=DE.EH$

In triangles $△ADEand△HGC$
$\angle A=\angle H(\because \text{alternate int. angles})$

$\angle AED=\angle ACB=\angle GCH(\because \text{corresponding and vertically opp. angles})$

So, $△ADE\sim △HGC$(AA property)

Now, in triangles $△ABC\text{and}△HFE\angle A=\angle H(\because \text{alternate int. angles})\angle ACB=\angle HEF(\because \text{alternate int. angles})$
So, $△ABC\sim △HFE$ (AA property)

we know, in similar triangles the corresponding sides are in the same ratio,so $\frac{AB}{HF}=\frac{BC}{FE}$
or $AB\times FE=BC\times HF$


In triangles $△ADE\text{and}△HFE\angle A=\angle H(\because \text{alternate angles})\angle AED=\angle HEF(\because \text{vertically opp. angles})$
So, $△ADE\sim △HFE$ (AA property)

we know, in similar triangles the corresponding sides are in the same ratio,so $\frac{AE}{EH}=\frac{DE}{EF}$
or $AE\times FE=DE\times EH$

So, option$\left(a\right)$, $\left(b\right)$ and $\left(c\right)$ correct.