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Mathematics

In the adjoining quadrilateral ABCD, AB is the longest side and DC is the shortest side. Prove that :

(i) ∠C > ∠A

(ii) ∠D > ∠B

In the adjoining quadrilateral ABCD, AB is the longest side and DC is the shortest side. Prove that :R.S. Aggarwal Mathematics Solutions ICSE Class 9.

Triangles

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Answer

Solve the following system of equations by using the method of cross multiplication: R.S. Aggarwal Mathematics Solutions ICSE Class 9.

(i) Given,

In quadrilateral ABCD,

AB is the longest sides and DC is the shortest side.

Join BD and AC.

In △ABC,

⇒ AB > BC

∴ ∠1 > ∠2 …..(1) [If two sides of a triangle are unequal, the greater side has the greater angle opposite to it]

In △ADC,

⇒ AD > DC

∴ ∠7 > ∠4 …..(2) [If two sides of a triangle are unequal, the greater side has the greater angle opposite to it]

Adding eq.(1) and (2), we get:

⇒ ∠1 + ∠7 > ∠2 + ∠4

⇒ ∠C > ∠A

Hence, proved that ∠C > ∠A.

(ii) In △ABD,

⇒ AB > AD

∴ ∠5 > ∠6 …..(1) [If two sides of a triangle are unequal, the greater side has the greater angle opposite to it]

In △BDC,

⇒ BC > CD

∴ ∠3 > ∠8 …..(2) [If two sides of a triangle are unequal, the greater side has the greater angle opposite to it]

Adding eq.(1) and (2), we get:

⇒ ∠5 + ∠3 > ∠6 + ∠8

⇒ ∠D > ∠B

Hence, proved that ∠D > ∠B.

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