Question

In Fig. 16.137, O and O’ are centres of two circles intersecting at B and C. ACD is a straight line, find x.

Answer 1

The value of x is 130°.

Step-by-step explanation:

O and O' are centres of two circles intersecting at B and C.

ACD is a straight line.

By degree measure theorem,

The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.

$\angle A O B=2 \angle A C B$

$\Rightarrow 130^{\circ}=2 \angle A C B$

$\Rightarrow \angle A C B=\frac{130^{\circ}}{2}$

⇒ ∠ACB = 65°

$\therefore \angle A C B+\angle B C D=180^{\circ}$

$\Rightarrow 65^{\circ}+\angle B C D=180^{\circ}$

$\Rightarrow \angle B C D=180^{\circ}-65^{\circ}$

⇒ ∠BCD = 115°

By degree measure theorem,

$\text { Reflex } \angle B O D=2 \angle B C A$

$\Rightarrow \text { Reflex } \angle B O D=2 \times 115^{\circ}$

$\Rightarrow \text { Reflex } \angle B O D=230^{\circ}$

Complete angle = 360°

$\text { reflex } \angle B O D+\angle B O D=360^{\circ}$

$230^{\circ}+x=360^{\circ}$

$x=360^{\circ}-230^{\circ}$

$x=130^{\circ}$

The value of x is 130°.

To learn more...

1. Two circles intersect at two points B and C. Through B, two line segments ABD and PBQ are drawn to intersect the circles at A, D and P, Q respectively (see the given figure). Prove that ∠ACP = ∠QCD.

https://brainly.in/question/1427972

2.  If a line intersects two concentric circles (circles with the same centre) with centre O at A, B, C and D, prove that AB = CD (see figure 10.25).

https://brainly.in/question/1427965

Answer 2

The value of x is 130°

Step-by-step explanation:

By using degree measure theorem,

∠AOB = 2∠ACB

From diagram, we get,

130° = 2∠ACB

∠ACB = 130°/2

∴ ∠ACB = 65°

Now,

∠ACB + ∠BCD = 180°

On substituting the value of ∠ACB, we get,

65° + ∠BCD = 180°

∠BCD = 180° - 65°

∴ ∠BCD = 115°

Again by using degree measure theorem,

Reflex ∠BOD = 2 ∠BCA

On substituting the value of ∠BCA, we get,

Reflex ∠BOD = 2 × 115° = 230°

Now,

Reflex ∠BOD + ∠BOD = 360°

230° + x = 360°

x = 360° - 230°

∴ x = 130° = ∠BOD