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Question 1 : Three angles of a quadrilateral are respectively equal to 1100, 500 and 400. Find its fourth angle.

Three angles of a quadrilateral are 1100,500 and 400

Let the fourth angle be ‘x’

We know, sum of all angles of a quadrilateral = 3600

1100 + 500 + 400 +x0 = 3600

x = 3600 – 2000

x = 1600

Therefore, the required fourth angle is 1600.

Question 2 : In a quadrilateral ABCD, the angles A, B, C and D are in the ratio of 1:2:4:5. Find the measure of each angles of the quadrilateral.

Let the angles of the quadrilaterals are A = x, B = 2x, C = 4xand D = 5x

We know, sum of all angles of a quadrilateral = 3600

A + B + C + D = 3600

x + 2x + 4x + 5x = 3600

12x = 3600

x = 3600/12 = 300

Therefore,

A = x = 300

B = 2x = 600

C = 4x = 1200

D = 5x = 1500

Question 3 : In a quadrilateral ABCD, CO and DO are the bisectors of ∠C and ∠D respectively. Prove that ∠COD = 1/2 (∠A + ∠B).

In ΔDOC,

CDO + COD + DCO = 1800 [Anglesum property of a triangle]

or 1/2CDA + COD + 1/2DCB = 1800

COD =1800 – 1/2(CDA + DCB) …..(i)

Also

We know, sum of all angles of a quadrilateral = 3600

CDA + DCB = 3600 –(DAB + CBA) ……(ii)

Substituting (ii) in (i)

COD =1800 – 1/2{3600 – (DAB + CBA) }

We can also write, DAB = A and CBA = B

COD =180− 180+1/2(A + B))

COD =1/2(A + B)

Hence Proved.

Question 4 : The angles of a quadrilateral are in the ratio 3:5:9:13. Find all the angles of the quadrilateral.

The angles of a quadrilateral are 3x, 5x, 9x and 13xrespectively.

We know, sum of all interior angles of a quadrilateral = 3600

Therefore, 3x + 5x + 9x + 13x = 3600

30x = 3600

or x = 120

Hence, angles measures are

3x = 3(12) = 360

5x = 5(12) = 600

9x = 9(12) = 1080

13x = 13(12) = 1560

krishan