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 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.
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 ∠C and ∠D respectively. Prove that ∠COD = 1/2 (∠A + ∠B).
Answer - 3 : -
In ΔDOC,
∠CDO + ∠COD + ∠DCO = 1800 [Anglesum property of a triangle]
or 1/2∠CDA + ∠COD + 1/2∠DCB = 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 =1800 − 1800 +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.
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