Question

the circle below has center m. suppose that ( moverarc{kl} = 92^circ ). find the following.
(a) ( mangle kjl = square^circ )
(b) ( mangle kml = square^circ )

Explanation:

Step1: Recall Central and Inscribed Angles

• Central angle theorem: The measure of a central angle is equal to the measure of its intercepted arc. So for \(\angle KML\) (central angle) and arc \(\overarc{KL}\), \(m\angle KML = m\overarc{KL}\).
• Inscribed angle theorem: The measure of an inscribed angle is half the measure of its intercepted arc. So for \(\angle KJL\) (inscribed angle) and arc \(\overarc{KL}\), \(m\angle KJL=\frac{1}{2}m\overarc{KL}\).

Step2: Solve for \(m\angle KJL\)

Given \(m\overarc{KL} = 92^\circ\), using the inscribed angle theorem:
\(m\angle KJL=\frac{1}{2}\times92^\circ = 46^\circ\).

Step3: Solve for \(m\angle KML\)

Using the central angle theorem, since \(\angle KML\) is a central angle intercepting arc \(\overarc{KL}\):
\(m\angle KML = m\overarc{KL}=92^\circ\).

Answer:

(a) \(m\angle KJL = \boldsymbol{46}\)
(b) \(m\angle KML = \boldsymbol{92}\)