Question

Prove that
"If two triangles are equiangular then their corresponding sides are in proportion"

Answer 1

Data : In $△ABC$ and $△DEF$
$\angle BAC=\angle EDF$
$\angle ABC=\angle DEF$
To prove : $\frac{AB}{DE}=\frac{BC}{EF}=\frac{CA}{FD}$
Construction : Mark points $G$ and $H$ on $AB$ and $AC$ such that $AG=DE$ and $AH=DF$. Join $G$ and $H$.
Proof : In $△AGH$ and $△DEF$
$AG=DE\because Construction$
$\angle GAH=\angle EDF\because Data$
$AH=DF\therefore Construction$
$\therefore △AGH\cong △DEF\because SAS$
$\angle AGH=\angle DEF$
But, $\angle ABC=\angle DEF$
$\Rightarrow \angle AGH=\angle ABC$
$\therefore GH\parallel BC$.
In $△ABC=\frac{AB}{AG}=\frac{BC}{GH}=\frac{CA}{HA}$
Hence $\frac{AB}{DE}=\frac{BC}{EF}=\frac{CA}{FD}$.