AB and CD are respectively the smallest and longest sides of a quadrilateral ABCD (see Figure). Show that A > C and B > D.
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12/07/2021 12:26 pm
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AB and CD are respectively the smallest and longest sides of a quadrilateral ABCD (see Figure). Show that A > C and B > D.
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12/07/2021 12:28 pm
In ΔABD, we see that
AB < AD < BD
So, ADB < ABD ...... (i) (Since angle opposite to longer side is always larger)
Now, in ΔBCD,
BC < DC < BD
Hence, it can be concluded that
BDC < CBD — (ii)
Now, by adding equation (i) and equation (ii) we get,
ADB + BDC < ABD + CBD
ADC < ABC
B > D
Similarly, In triangle ABC,
ACB < BAC — (iii) (Since the angle opposite to the longer side is always larger)
Now, In ΔADC,
DCA < DAC — (iv)
By adding equation (iii) and equation (iv) we get,
ACB + DCA < BAC+DAC
⇒ BCD < BAD
∴ A > C
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