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Answer:
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
