Question

applying theorems
what are the measures of \\(\overarc{jk}\\) and \\(\angle kij\\)?
measure of \\(\overarc{jk} = \boxed{1}
measure of \\(\angle kij = \boxed{}
(image of a circle with points k, j, i on the circumference, angle at k is 70°, arc from k to i is 116°)

Explanation:

Step1: Find the measure of arc \( JK \)

The sum of the measures of the arcs of a circle is \( 360^{\circ} \). Let the measure of arc \( JK \) be \( x \).
We know that the inscribed - angle theorem states that the measure of an inscribed angle is half the measure of its intercepted arc.
The inscribed angle \( \angle KIJ = 70^{\circ} \), and it intercepts arc \( JK \). By the inscribed - angle theorem \( \angle KIJ=\frac{1}{2}\text{measure of arc }JK\).
So, measure of arc \( JK = 2\times\angle KIJ\).
Since \( \angle KIJ = 70^{\circ} \), measure of arc \( JK=140^{\circ}\).

Step2: Find the measure of \( \angle KLJ \)

The measure of an inscribed angle is half the measure of its intercepted arc.
The arc \( KI \) has a measure of \( 116^{\circ} \).
The angle \( \angle KLJ \) is an inscribed angle that intercepts arc \( KI \).
By the inscribed - angle theorem \( \angle KLJ=\frac{1}{2}\text{measure of arc }KI\).
Since measure of arc \( KI = 116^{\circ} \), then \( \angle KLJ=\frac{116^{\circ}}{2}=58^{\circ}\).

Answer:

Measure of \( \overarc{JK}=140^{\circ} \)
Measure of \( \angle KLJ = 58^{\circ} \)