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.


Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts
Dedicated counsellor for each student
24X7 Doubt Resolution
Daily Report Card
Detailed Performance Evaluation
caricature
view all courses
RELATED QUESTIONS :

In ΔABC, AC > West Bengal - Mathematics

In ΔABC, AD is peWest Bengal - Mathematics

In ΔPQR, PQ > West Bengal - Mathematics

In a quadrilateraWest Bengal - Mathematics

Pallabi and SirajWest Bengal - Mathematics

Let’s prove that West Bengal - Mathematics

In Δ<span lang="EWest Bengal - Mathematics

Let’s write the vWest Bengal - Mathematics

Let's measure theWest Bengal - Mathematics

Two straight lineWest Bengal - Mathematics