Question

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

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

Answer 1

its c

Explanation:

Answer 2

The sum of the measures of each pair of vertical angles is equal

to the sum of the measures of the intercepted arcs

Explanation:

* Lets study this rule :Angles of Intersecting Chords Theorem.

- If two chords intersect inside a circle, then the measure of the

angle formed is one half the sum of the measure of the arcs

intercepted by the angle and its vertical angle.

Ex: Look to the attached figure:

- QS and PR are 2 chords intersect each other inside the circle

- ∠1 subtended by arc PQ , ∠3 subtended by the arc RS

- ∠2 subtended by arc RQ , ∠4 subtended by the arc PS

∴ m∠1 = [measure of arc PQ + measure of arc RS]/2

∴ m∠2 = [measure of arc QR + measure of arc PS]/2

∴ 2m∠1 = measure of arc PQ + measure of arc RS

∴ 2m∠2 = measure of arc QR + measure of arc PS

∵ m∠1 = m∠3 ⇒ vertically opposite angles

∵ m∠2 = m∠4 ⇒ vertically opposite angles

∴ m∠1 + m∠3 = 2m∠1

∴ m∠2 + m∠4 = 2m∠2

∴ m∠1 + m∠3 = measure of arc PQ + measure of arc RS

∴ m∠2 + m∠4 = measure of arc QR + measure of arc PS

* From all above

- The sum of the measures of each pair of vertical angles is equal

to the sum of the measures of the intercepted arcs