Question

Prove that if in two triangles, corresponding angles are equal then their corresponding sides are in the same ratio and hence the two triangles are similar.

Answer 1

$Two△les△ABC\&△DEFsuchthat$

$\angle A=\angle D,\angle B=\angle E\&\angle C=\angle F$

$DrawPandQonDE\&DFsuchthatDP=AB$

$\&DQ=ACresp.JoinPQ$

$In△ABC\&△DPQAB=DP\left(byconstruction\right)$

$\angle A=\angle D\left(given\right)$

$AC=DQ\left(byconstruction\right)$

$⟹△ABC\cong △DPQ\left(bySAScriterion\right)$

$\angle B=\angle P\left\{byCPCT\right\}$

$\angle B=\angle E\left(given\right)implies\angle P=\angle E$

$ForlinesPQandEFwithtransversalPE,$

$\angle P\&\angle Earecorrespondinganglesandthey$

$areequal.Hence,PQ||EF.$