State True or False.

CD and GH are respectively the bisectors of ∠ACB and ∠ EGF such that D and H lie on sides AB and FE of ∆ ABC and ∆ EFG respectively.

If $Δ ABC ∼ Δ FEG$ then $C D G H = B C F G$ .

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True

False

answer is B.

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Detailed Solution

Given that CD and GH are respectively the bisectors of ∠ACB and ∠ EGF such that D and H lie on sides AB and FE of ∆ ABC and ∆ EFG respectively.
Question Image

Also, given that

Δ ABC ∼ Δ FEG

.
If two triangles are similar, then their corresponding angles are equal.
So,

∠ A = ∠ F, ∠ B = ∠ E and ∠ ACB = ∠ FGE .

In ΔDCB and Δ HGE,

∠ B = ∠ E

[proved above]

∠ DCB = ∠ HGE

[bisected angles of ∠ACB and ∠EGF]
So,

Δ DCB ~ Δ HGE

by AA similarity criteria.
We know that if two triangles are similar, then the ratios of their corresponding sides are in proportion.
Using the property of similar triangles,