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.

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

to the sum of the measures of the intercepted arcs

Step-by-step 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

∠1≅∠2 by the alternate exterior angles theorem.

step-by-step explanation:

given, a ∥ b and ∠1 ≅ ∠3 .we have to prove that e ∥ f

we know that ∠1≅∠3 and that a || b because they are given. we see that by the alternate exterior angles theorem. therefore, ∠2≅∠3 by the transitive property. so, we can conclude that e || f by the converse alternate exterior angles theorem.

we have to fill the missing statement.

transitivity property states that if a = b and b = c, then a = c.

now, given ∠1≅∠3 and by transitivity property ∠2≅∠3 .

hence, to apply transitivity property one angle must be common which is not in result after applying this property which is ∠1.

the only options in which ∠1 is present are ∠1 and ∠2, ∠1 and ∠4

∠1 and ∠4 is not possible ∵ after applying transitivity we didn't get ∠4.

hence, the missing statement is ∠1≅∠2.

so, ∠1≅∠2 by the alternate exterior angles theorem.