. If two triangles are equiangular, then their corresponding sides are in proportion . Prove.
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Answer:
According to Greek mathematician Thales, “The ratio of any two corresponding sides in two equiangular triangles is always the same.” ... If in two triangles, sides of one triangle are proportional to the sides of the other triangle, then their corresponding angles are equal and hence the two triangles are similar
Step-by-step explanation:
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Given: In ΔABC, DE is parallel to BC
Line DE intersects sides AB and AC in points D and E respectively.
To Prove: ADBD=AECE
Construction: Draw EF ⟂ AD and DG⟂ AE and join the segments BE and CD.
Proof:
Area of Triangle= ½ × base× height
In ΔADE and ΔBDE,
Ar(ADE)Ar(DBE)=12×AD×EF12×DB×EF=ADDB(1)
In ΔADE and ΔCDE,
Ar(ADE)Ar(ECD)=12×AE×DG12×EC×DG=AEEC(2)
Note that ΔDBE and ΔECD have a common base DE and lie between the same parallels DE and BC. Also, we know that triangles having the same base and lying between the same parallels are equal in area.
So, we can say that
Ar(ΔDBE)=Ar(ΔECD)
Therefore,
A(ΔADE)A(ΔBDE)=A(ΔADE)A(ΔCDE)
Therefore,
ADBD=AECE
Hence Proved.
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