Question

applying theorems
what are the measures of \\(\overarc{jk}\\) and \\(\angle klj\\)
?
measure of \\(\overarc{jk} = \square^\circ\\)
measure of \\(\angle klj = \square^\circ\\)
feedback
the measure of \\(\overarc{ji}\\) is twice the measure of \\(\angle jki\\).
there are \\(360^\circ\\) in a circle.

Explanation:

Step1: Use the inscribed - angle theorem

The measure of an inscribed angle is half the measure of its intercepted arc. Given that \(\angle JKI = 70^{\circ}\), and the measure of an inscribed angle \(\angle JKI=\frac{1}{2}\text{measure of }\overset{\frown}{JI}\). So, \(\text{measure of }\overset{\frown}{JI} = 2\times70^{\circ}=140^{\circ}\).
Since the sum of the measures of arcs in a circle is \(360^{\circ}\), and we know one arc is \(116^{\circ}\) and another is \(140^{\circ}\). Let \(\text{measure of }\overset{\frown}{JK}=x\). Then \(x + 116^{\circ}+140^{\circ}=360^{\circ}\).

$$x=360^{\circ}-(116^{\circ} + 140^{\circ})=360^{\circ}-256^{\circ}=104^{\circ}$$

Step2: Use the inscribed - angle theorem again

The measure of an inscribed angle \(\angle KIJ\) is half the measure of its intercepted arc \(\overset{\frown}{JK}\). So, \(\text{measure of }\angle KIJ=\frac{1}{2}\text{measure of }\overset{\frown}{JK}\).
Since \(\text{measure of }\overset{\frown}{JK} = 104^{\circ}\), then \(\text{measure of }\angle KIJ=\frac{1}{2}\times104^{\circ}=52^{\circ}\)

Answer:

Measure of \(\overset{\frown}{JK}=104^{\circ}\)
Measure of \(\angle KIJ = 52^{\circ}\)