Question

$\text{AB and CD are respectively the smallest and largest sides of a quadrilateral ABCD.}\text{Show that}\angle A>\angle Cand\angle B>\angle D.$

Answer 1

ANSWER:
Given: In quadrilateral ABCD, AB and CD are respectively the smallest and largest sides.
To prove:
(i) ∠ A > ∠ C
(ii) ∠ B > ∠ DConstruction: Join AC .
Proof:
In ∆ABC,∵ BC>AB Given, AB is the smallest side∴ ∠1>∠2
In ∆ADC,∵ CD>AD Given, CD is the largest side∴ ∠3>∠4
...1
...2
Adding 1 and 2, we get∠1+∠3>∠2+∠4∴ ∠A>∠C
(ii)
Construction: Join BD.
Proof:
In ∆ABD,∵ AD>AB Given, AB is the smallest side.∴ ∠5>∠6 ...3
In ∆CBD,∵ CD>BC Given, CD is the greatest side.∴ ∠7>∠8 ...4
Adding 3 and 4, we get∠5+∠7>∠6+∠8∴∠B>∠D