Question

From the following data, state whether $△$ABC is similar to $△$DEF or not.
(a) $\angle A={70}^o,\angle B={80}^o,\angle D={70}^o,\angle F={30}^o$
(b) $AB=8cm,BC=9cm,CA=15cm,DE=4cm,EF=3cm,FD=5cm$.

Answer 1

Consider the two figure, $△ABC$ and $△DEF$,

(i) Given $\angle A={70}^o$ and $\angle B={80}^o$

By Angle property, the sum of interior angles of triangle = ${180}^o$

$\Rightarrow \angle A+\angle B+\angle C={180}^o$

$\therefore 70+80+\angle C={180}^o$

$\therefore \angle C={30}^o$

Consider $△DEF$

$\angle D+\angle E+\angle F={180}^o$

$70+\angle E+30={180}^o$

$\angle E={80}^o$

Compare both $△ABC$ and $△DEF$,

$\angle A=\angle D={70}^o;\angle B=\angle E={80}^o;\angle C=\angle F={30}^o$

$\therefore △ABC\cong △DEF$ ....... [AAA Criteria]

(ii) $\frac{DE}{AB}=\frac{4}{8}=\frac{1}{2};\frac{DF}{AC}=\frac{5}{15}=\frac{1}{3}$

$\frac{EF}{BC}=\frac{3}{9}=\frac{1}{3}$

$\therefore AB=2DE$

$AC=3DF$

$CA=3EF$

$\therefore △ABC\cong △DEF$ ....... [by SSS criteria]