RD Chapter 14 Quadrilaterals Ex 14.1 Solutions
Question - 1 : - Three angles of a quadrilateral are respectively equal to 1100, 500 and 400. Find its fourth angle.
Answer - 1 : -
Three angles of a quadrilateral are 1100,500┬аand 400
Let the fourth angle be тАШxтАЩ
We know, sum of all angles of a quadrilateral = 3600
1100┬а+ 500┬а+ 400┬а+x0┬а= 3600
тЗТ x = 3600┬атАУ 2000
тЗТx = 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.
Answer - 2 : -
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 тИаC and тИаD respectively. Prove that тИаCOD = 1/2 (тИаA + тИаB).
Answer - 3 : -
In ╬ФDOC,
тИаCDO + тИаCOD + тИаDCO = 1800┬а[Anglesum property of a triangle]
or 1/2тИаCDA + тИаCOD + 1/2тИаDCB = 1800
┬атИаCOD =1800┬атАУ 1/2(тИаCDA + тИаDCB) тАж..(i)
Also
We know, sum of all angles of a quadrilateral = 3600
тИаCDA + тИаDCB = 3600┬атАУ(тИаDAB + тИаCBA) тАжтАж(ii)
Substituting (ii) in (i)
тИаCOD =1800┬атАУ 1/2{3600┬атАУ (тИаDAB + тИаCBA) }
We can also write, тИаDAB = тИаA and тИаCBA = тИаB
тИаCOD =1800┬атИТ 1800┬а+1/2(тИаA + тИаB))
тИаCOD =1/2(тИаA + тИаB)
Hence Proved.
Question - 4 : - The angles of a quadrilateral are in the ratio 3:5:9:13. Find all the angles of the quadrilateral.
Answer - 4 : -
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