prove that if two triangles are equiangler then their corresponding sides are proportional
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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.
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Answer:
Prove that
"If two triangles are equiangular then their corresponding sides are proportional
ANSWER
Data : In △ABC and △DEF
∠BAC=∠EDF
∠ABC=∠DEF
To prove :
DE
AB
=
EF
BC
=
FD
CA
Construction : Mark points G and H on AB and AC such that AG=DE and AH=DF. Join G and H.
Proof : In △AGH and △DEF
AG=DE∵Construction
∠GAH=∠EDF∵Data
AH=DF∴Construction
∴△AGH≅△DEF∵SAS
∠AGH=∠DEF
But, ∠ABC=∠DEF
⇒∠AGH=∠ABC
∴GH∥BC.
In △ABC=
AG
AB
=
GH
BC
=
HA
CA
Hence
DE
AB
=
EF
BC
hope it helps