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
Ā angle formed is one half the sum of the measure of the arcs
Ā 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
- ā 1 subtended by arc PQ , ā 3 subtended by the arc RS
- ā 2 subtended by arc RQ , ā 4 subtended by the arc PS
ā“ mā 1 = [measure of arc PQ + measure of arc RS]/2
ā“ mā 2 = [measure of arc QR + measure of arc PS]/2
ā“ 2mā 1 = measure of arc PQ + measure of arc RS
ā“ 2mā 2 = measure of arc QR + measure of arc PS
āµ mā 1 = mā 3 ā vertically opposite angles
āµ mā 2 = mā 4 ā vertically opposite angles
ā“ mā 1 + mā 3 = 2mā 1
ā“ mā 2 + mā 4 = 2mā 2
ā“ mā 1 + mā 3 = measure of arc PQ + measure of arc RS
ā“ mā 2 + mā 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
Ā to the sum of the measures of the intercepted arcs
