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Inequality in a Triangle

Theorem 1: If two sides of a triangle are unequal, the longer side has greater angle opposite to it.
Given: In
ΔABC, AC > AB
To Prove:
ABC > ACB
Construction: Mark a point D on AC, such that AB = AD and join BD.

Proof:
In ΔABD,
since AB = AD ... (by construction)

...      1 = 2 ... (Angles opposite to equal sides)
Now
2 is the exterior angle of ΔBCD.
... 2 = DCB + CBD
...
2 > DCB
or
2 > ACB
... 1 > ACB
... 1 + DBC > ACB
Hence
ABC > ACB
 



 




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