In given figure AB and CD are respectively the smallest and longest sides of a quadrilateral ABCD. Show that ∠A∠C and ∠B∠D.

Question

In given figure AB and CD are respectively the smallest and longest sides of a quadrilateral ABCD. Show that ∠A>∠C and ∠B>∠D.

Filo · Accepted Answer

ABCD is a quadrilateral such that AB is its smallest side and CD is its largest side. Join AC and BD. Since AB is the smallest side of quadrilateral ABCD. ∴ In △ ABC, we have BC>AB⇒   ∠8>∠3 ... (1) [Since angle opposite to longer side is greater]Since CD is the longest side of quadrilateral ABCD. Now, In  △ ACD, we have CD>AD ⇒    ∠7>∠4 ... (2) [Since angle opposite to longer side is greater]Adding (1) and (2), we get ∠8+∠7>∠3+∠4⇒   ∠A>∠CAgain, in △ ABD, we have AD>AB [Since AB is the shortest side]⇒   ∠1>∠6... (3) In △ BCD, we have CD>BC [Since CD is the longest side]⇒ ∠2>∠5 ... (4) Adding (3) and (4), we get ∠1+∠2>∠5+∠6 ⇒∠B>∠DThus , ∠A>∠C and ∠B>∠D.   [henceproved]

Question Text	In given figure $A B$ and $C D$ are respectively the smallest and longest sides of a quadrilateral $A BC D$ . Show that $∠ A > ∠ C$ and $∠ B > ∠ D$ .
Answer Type	Text solution:1
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In given figure $A B$ and $C D$ are respectively the smallest and longest sides of a quadrilateral $A BC D$ . Show that $∠ A > ∠ C$ and $∠ B > ∠ D$ .

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