Question

HURRY!
In the diagram of circle O, what is the measure of angle ABC?

Answer 1

Measure of Angle ABC is 30°

Explanation:

Given: Circle with centre O and Tangents AB & BC.

To find: m∠ABC.

Construction: Join Radius OA & OC. (New Figure is attached)

Given Value of Arc AC is 150°. It means 150° is measure of angle by which that arc is suspended.

⇒ m∠AOC = 150°.

m∠BAO = m∠BCO = 90° ( because Tangent and Radius are perpendicular to each other at point of contact i.e., A & B )

ABCO is a Quadrilateral.

∴ using Angle Sum Property of Quadrilateral which states that sum of all interior angles of quadrilateral is 360° , we get

∠BAO + ∠BCO + ∠ABC + ∠AOC = 360

⇒ 90 + 90 + ∠ABC + 150 = 360

⇒ ∠ABC + 330 = 360

⇒ ∠ABC = 30°

Measure of Angle ABC is 30° .

Answer 2

Answer: ∠ABC=30°

Given: A circle with center O. Two tangents AB and BC are making angle ABC with arc at circle 150°.

To find : ∠ABC

Solution: We know that the angle formed by the intersection of two tangents outside the circle equals to the half of difference of the intercepted arcs.[major arc and minor arc]

From the figure ,Minor arc = 150°

Major arc = 360 - 150 = 210°

Therefore,∠ABC =
`(210^(\circ)-150^(\circ))/(2)=(60^(\circ))/(2)=30^(\circ)`

Thus ∠ABC=30°.