10) The angles A, B, C and D in order in a cyclic quadrilateral are (2x+y),
(2(x+y)),(3x+2y)o, and (4x-2y)o. Find their measures in the same order.
A) 70 ,110 ,80, 100
B) 70 ,80,110 ,100
C) 70 ,80, 100 ,110.
D) 80, 100 ,110 ,70
Answers
Answer:
D) 80, 100 ,110 ,70
Hope it will help you
Answer:
The four angles are 70°, 80°,110°,100°.
Hence option(B) is correct answer.
Step-by-step explanation:
Cyclic Quadrilateral:
- A cyclic quadrilateral is a quadrilateral which has all four vertices lying on the circle.
- Sometimes it is also called as Inscribed Quadrilateral.
- The circle which consists of all the vertices of any polygon on its circumference is known as the circumcircle.
Properties of Cyclic Quadrilateral:
- The sum of opposite angles in a cyclic quadrilateral is Supplementary.
- Let ∠A, ∠B, ∠C , and ∠D are the four angles of an inscribed quadrilateral then
- ∠A + ∠C = 180° and ∠B + ∠D = 180°
Given angles of a cyclic quadrilateral are
(2x+y), 2(x+y), 3x+2y, and 4x-2y
Let us suppose the angles as
∠A = (2x+y), ∠B = 2(x+y), ∠C = 3x+2y, and ∠D = 4x-2y
According to the property of Cyclic quadrilateral,
Sum of opposite angles is equal to 180°.
∠A + ∠C = 180° and ∠B + ∠D = 180°
Consider, ∠A + ∠C = 180°
2x+y+3x+2y = 180°
2x+3x+y+2y = 180°
5x+3y = 180° ------------(i)
Consider, ∠B + ∠D = 180°
2(x+y)+4x-2y = 180°
2x+2y+4x-2y = 180°
6x+0 = 180°
6x = 180°
x = 180°/6
x = 30
now substitute the value of x in equation (i)
5x+3y = 180°
5(30)+3y = 180°
150 +3y = 180°
3y = 180°- 150°
3y = 30
y = 10
Now substitute the values of x = 30 and y = 10 in all the four angles.
∠A = (2x+y) = 2(30)+10 = 70°
∠B = 2(x+y) = 2(30+10) = 80°
∠C = 3x+2y = 3(30)+2(10) = 110°
∠D = 4x-2y = 4(30) -2(10) = 100°
Hence, the four angles are 70°, 80°,110°,100°.
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