Question

on circle o below, the measure of \\(\overarc{fj}\\) is \\(84^\circ\\). the measure of \\(\overarc{gh}\\) is \\(76^\circ\\). what is the measure of \\(\angle hkj\\)?

Explanation:

Step1: Recall the theorem for angle formed by two chords

When two chords intersect inside a circle, the measure of the angle formed is equal to half the sum of the measures of the intercepted arcs. So, \( m\angle HKJ=\frac{1}{2}(m\widehat{FJ} + m\widehat{GH}) \).

Step2: Substitute the given arc measures

We know \( m\widehat{FJ} = 84^\circ \) and \( m\widehat{GH}=76^\circ \). Substitute these values into the formula: \( m\angle HKJ=\frac{1}{2}(84^\circ + 76^\circ) \).

Step3: Calculate the sum of the arcs

First, add the measures of the arcs: \( 84^\circ+76^\circ = 160^\circ \).

Step4: Find the measure of the angle

Then, take half of the sum: \( \frac{1}{2}\times160^\circ = 80^\circ \).

Answer:

\( 80 \)