Q. 55.0( 1 Vote )

# In a quadrilatera

Answer :

Given: a quadrilateral such that the greatest and the smallest sides are opposite to each other

To prove: the adjacent angle to the greatest side is smaller than its opposite side

Let ABCD be the quadrilateral, then according to the given criteria the figure is as shown below,

Let us draw two diagonals BD and AC as shown in the figure.

In ΔABD,

AB < AD < BD (as AB is the smallest side of the quadrilateral)

So, ∠ADB < ∠ABD - - - - - - - - (1)

(angle opposite to smaller side is smaller)

In ΔBCD

BC < DC < BD (As CD is the longest side in the quadrilateral, given)

So, ∠BDC < ∠CBD - - - - - - - - - (2)

Adding equations (1) and (2)

∠ADB + ∠BDC < ∠ABD + ∠CBD

Or, ∠ADC < ∠ABC………..(i)

Similarly, in ΔABC

AB < BC < AC

∠ACB < ∠BAC - - - - - - - - - (3)

in ΔADC

AD < AC

∠DCA < ∠DAC - - - - - - - - (4)

Adding equations (3) and (4)

∠ACB + ∠DCA < ∠DAC + ∠BAC

Or, ∠BCD < ∠BAD…….(ii)

Hence from equation (i) and (ii), we can conclude that the adjacent angle to the greatest side is smaller than its opposite side.

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