We can use a few more theorems to find the measures of arcs and central angles of circles.  Let’s begin by stating a few theorems:

THEOREM:  The measure of a central angle is equal to the measure of the arc it intersects.

THEOREM:  The measure of a major arc (an arc greater than a semicircle) is equal to \(360^\circ \) minus the measure of the corresponding minor arc.

THEOREM:  Vertical angles are equal.

EXAMPLE:  Find the measure of the arc \(\widehat {IKH}\)

SOLUTION: \(\widehat {IKH}\) is a major arc, so, by a theorem above, its measure is \(360^\circ  - m\widehat {IH}\). Then we must find the measure of \(\widehat {IH}\).

Again, by a theorem above, we know that the measure of the central angle corresponding to \(\widehat {JK}\) must be \(70^\circ \). Now we can conclude, by the fact that straight angles measure \(180^\circ \), that the central angle corresponding to \(\widehat {IH}\) equals \(180 - 70 - 70 = 40\). That is, \(m\widehat {IH} = 40^\circ \).

So we can conclude that \(m\widehat {IKH} = 360^\circ  - 40^\circ  = 320^\circ \).

EXAMPLE:  Find \(m\angle JKI\)

SOLUTION:  Since \(\widehat {FKG} = 50^\circ \), we can conclude, by a theorem above, that \(\angle FKG = 50^\circ \). Then, by the vertical angle theorem, we know that \(\angle JKL = \angle FKG\). That is, \(\angle JKL = 50^\circ \).

Below you can download some free math worksheets and practice.