# CD and GH are respectively the bisectors of ∠ ACB and ∠ EGF in such a way that D and H lie on sides AB and FE of Δ ABC and Δ EFG respectively. If Δ ABC ~ Δ FEG, show that:(i) (ii) Δ DCB ~ Δ HGE(iii) Δ DCA ~ Δ HGF Given, Δ ABC Δ FEG …..eq(1)

corresponding angles of similar triangles

BAC = EFG ….eq(2)

And ABC = FEG …….eq(3)

ACB = FGE ACD = FGH and BCD = EGH ……eq(4)

Consider Δ ACD and Δ FGH

From eq(2) we have

DAC = HFG

From eq(4) we have

ACD = EGH

If the 2 angle of triangle are equal to the 2 angle of another triangle, then by angle sum property of triangle 3rd angle will also be equal.

by AAA similarity we have in two triangles if the angles are equal, then sides opposite to the equal angles are in the same ratio (or proportional) and hence the triangles are similar.

By Converse proportionality theorem Consider Δ DCB and Δ HGE

From eq(3) we have

DBC = HEG

From eq(4) we have

BCD = FGH

Also, BDC = EHG

Δ DCB ΔHGE

Hence proved.

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