Question

In each of the given pairs of triangles, find which pair of triangles are similar. State the similarity criterion and write the similarity relation in symbolic form.

(i)

(ii)

(iii)

(iv)

(v)

Answer 1

(i)

$\Delta ABCand\Delta PQR\angle A=\angle Q\angle B=\angle P\angle C=\angle R\Delta ABC\sim \Delta PQR\left(byAAAsimilarity\right)$

(ii)

$In\Delta ABCand\Delta EFD\angle A=\angle Dand\frac{AB}{FD}=\frac{3}{6}=\frac{1}{2}\frac{BC}{ED}=\frac{4.5}{9}=\frac{1}{2}Hence\frac{AB}{FD}=\frac{BC}{ED}=\frac{1}{2}So\Delta ABC\sim \Delta EFD\left(bySASsimilarity\right)$

(iii)

$In\Delta ABCand\Delta PQR\angle C=\angle Qand\frac{CA}{QR}=\frac{8}{6}=\frac{4}{3}\frac{CB}{QP}=\frac{6}{4.5}=\frac{4}{3}Hence\frac{CA}{QR}=\frac{CB}{QP}=\frac{4}{3}So\Delta ABC\sim \Delta PQR\left(bySASsimilarity\right)$

(iv)

Finding ratio of sides

$\frac{ED}{QR}=\frac{2.5}{5}=\frac{1}{2}\frac{EF}{PQ}=\frac{2}{4}=\frac{1}{2}\frac{DF}{PR}=\frac{3}{6}=\frac{1}{2}Hence\frac{ED}{QR}=\frac{EF}{PQ}=\frac{DF}{PR}DFPR=\frac{1}{2}So\Delta DEF\sim \Delta PQR\left(SSSsimilarity\right)$

(v)

$\Delta ABCand\Delta MNRAnglesum={180}^{o}In\Delta ABC,\angle A+\angle B+\angle C={180}^o\angle B={180}^o-({70}^o+{80}^o)={30}^{o}In\Delta MNR,\angle M+\angle R+\angle N={180}^o\angle R={180}^o-({30}^o+{80}^o)={70}^o\angle A=\angle M\angle B=\angle N\angle C=\angle R\Delta ABC\sim \Delta MNR\left(byAAAsimilarity\right)$