Two chords intersect at a point inside a circle that is not the center of the circle. which statement must be true?

a. the measures of each pair of opposite arcs sum to 180°.
b. each angle measure is equal to the measure of the arc it intercepts.
c. the sum of the measures of each pair of vertical angles is equal to the sum of the measures of the intercepted arcs.
d. each arc measure is equal to one-half the sum of the vertical angles that intercept the arc.

- C is the right option.

Explanation:-

Let O be the center of the circle and x be the measure of ∠AOC formed by intersection of two chords AB and CD.[as shown in the attachment]

Let the measure of major arc be 'a' and the measure of minor arc be 'b'.

We know that angles that are formed inside of a circle by two chords create four arcs on a circle. The measure of the angle is equal to half the sum of the intercepted arcs.

And ∠AOC=∠BOD [vertically opposite angles]

The sum of the measures of each pair of vertical angles

⇒The sum of the measures of each pair of vertical angles is equal to the sum of the measures of the intercepted arcs.