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Answer:  The correct option is (C) 120°.

Step-by-step explanation:  Given that Ac and BD are the diameters of the circle with center 'O'.

We are to find the measure of arc AB.

Since AC and BD are diameters of the circle, and diameters subtends an angle of 180° at the center, so we have

[tex]m\angle AOC=180^\circ,\\\\m\angle BOD = 180^\circ.[/tex]

From the figure, we get

[tex]m\angle AOC=180^\circ\\\\\Rightarrow m\angle AOB+x^\circ=180^\circ\\\\\Rightarrow m\angle AOB=180^\circ-x^\circ~~~~~~~~~~~~~~~~~~(i)[/tex]

Since the diameters AC and BD intersect at the center 'O', so ∠AOC and ∠BOD are vertically opposite angles. Hence, they must be equal.

So,

[tex]m\angle AOB=m\angle COD\\\\\Rightarrow 180^\circ-x^\circ=x^\circ+x^\circ~~~~~~~~~~~~~~~~~~~~~\textup{[using equation (i)]}\\\\\Rightarrow 3x^\circ=180^\circ\\\\\Rightarrow x^\circ=60^\circ.[/tex]

Therefore, from equation (i), we get

[tex]m\angle AOB=180^\circ-60^\circ\\\\\Rightarrow m\angle AOB=120^\circ\\\\\Rightarrow m~\textup{arc AB}=120^\circ.[/tex]

Thus, the measure of arc AB is 120°. Option (C) is correct.

You can use the fact that opposite angles made between two intersecting lines are of equal measurement.

The measure of arc AB in terms of angles is

[tex]Arc \: AB = 120^\circ[/tex]

What is the sum of angles in full rotation?

When you rotate a line and come back to that same position after a single full rotation, then that angle is measured as [tex]360^\circ[/tex](degrees) or [tex]2\pi^c[/tex](radians)

How to find the measure of arc AB?

Since the angle AOB is equal to angle BOC because of both of them being opposite angles made by intersecting lines, we get:

[tex]\angle AOB = \angle BOC = x^\circ + x^\circ = 2x^\circ[/tex]

Since the sum of angles for full rotation is of 360 degrees, thus

[tex]\angle AOB + \angle DOC + \angle AOC + \angle BOC = 360^\circ\\2x^\circ + 2x^\circ + x^\circ + x^\circ= 360^\circ\\6x^\circ = 360^\circ\\\\x = \dfrac{360}{6} = 60[/tex]

The measure of arc AB is same as the angle it subtends on center which is angle AOB which is 2x degrees, thus

[tex]Arc AB = 2x^\circ = 2 \times 60 = 120^\circ[/tex]

Thus,

The measure of arc AB in terms of angles is

[tex]Arc \: AB = 120^\circ[/tex]

Learn more about arcs and angles here:

https://brainly.com/question/12666182

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