(a) If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, then prove that the other two sides…

Correct Given, to prove, figure, construction Correct proof

Produce AD to E such that \(\displaystyle \mathrm{AD}=\mathrm{DE}\) and join EC Produce PM to N such that PM = MN and join NR \(\displaystyle \Delta \mathrm{ADB} \cong \triangle \mathrm{EDC} \therefore \mathrm{AB}=\mathrm{EC}\)

Similarly, \(\displaystyle \mathrm{PQ}=\mathrm{NR}\) Since, \(\displaystyle \frac{\mathrm{AB}}{\mathrm{PQ}}=\frac{\mathrm{AC}}{\mathrm{PR}}=\frac{\mathrm{AD}}{\mathrm{PM}} \Rightarrow \frac{\mathrm{EC}}{\mathrm{NR}}=\frac{\mathrm{AC}}{\mathrm{PR}}=\frac{\dfrac{\mathrm{AE}}{2}}{\dfrac{\mathrm{PN}}{2}} \therefore \triangle \mathrm{AEC} \sim \triangle \mathrm{PNR} \Rightarrow \angle 1=\angle 2\)

Similarly, \(\displaystyle \angle 3=\angle 4\) Hence \(\displaystyle \angle 1+\angle 3=\angle 2+\angle 4\) or \(\displaystyle \angle \mathrm{A}=\angle \mathrm{P}\) Also, \(\displaystyle \frac{\mathrm{AB}}{\mathrm{PQ}}=\frac{\mathrm{AC}}{\mathrm{PR}} \therefore \triangle \mathrm{ABC} \sim \triangle \mathrm{PQR}\)