∠ACB is inscribed in arc ACB of a circle with centre O. If ∠ACB = 65°, find m(arc ACB). 65° 130° 295° 230°

Explanation:_

An inscribed angle is an angle formed by two chords in a circle which have a common endpoint on the circle. The measure of an inscribed angle is half of the measure of the arc intercepted by it.

In this problem, we are given that m∠ACB is half of the measure of the arc AB, and we need to find the measure of the arc ACB.

Using the inscribed angle theorem, we know that:

m∠ACB = 1/2 m(arc AB)

We are given that m∠ACB = 65°. Substituting this value in the above equation, we get:

1/2 m(arc AB) = 65°

Simplifying this equation, we get:

m(arc AB) = 2 × 65° = 130°

Now, we need to find the measure of the arc ACB. We know that the sum of the measures of two arcs which are opposite to each other is 360°. Therefore:

m(arc ACB) = 360° – m(arc AB)

Substituting the value of m(arc AB), we get:

m(arc ACB) = 360° – 130° = 230°

Therefore, the measure of the arc ACB is 230°.