Question

if ( mangle mpl = 63^{circ} ), find each measure

1. ( mwidehat{nj}=square^{circ} )
2. ( mwidehat{jlk}=square^{circ} )
3. ( mwidehat{jl}=square^{circ} )

Explanation:

Step1: Find \(m\overarc{NJ}\)

Since \(\angle MPL = 63^{\circ}\), and vertical angles are equal. The central angle for \(\overarc{NJ}\) is equal to \(\angle MPL\). So \(m\overarc{NJ}=63^{\circ}\)

Step2: Find \(m\overarc{JLK}\)

The total degrees in a circle is \(360^{\circ}\). We know that \(\angle JPL = 90^{\circ}\) (right - angle). \(m\overarc{JLK}\) is composed of \(\overarc{JL}\) and \(\overarc{LK}\). First, find \(m\overarc{JL}\): Since \(\angle JPL = 90^{\circ}\), \(m\overarc{JL}=90^{\circ}\). The central angle for \(\overarc{LK}\) is equal to \(\angle MPL = 63^{\circ}\). \(m\overarc{JLK}=m\overarc{JL}+m\overarc{LK}+m\overarc{KJ}\). But another way: \(m\overarc{JLK}=360^{\circ}-m\overarc{JN}-m\overarc{NL}\). Since \(\angle NPL=\angle MPL = 63^{\circ}\) (vertical angles), \(m\overarc{NL}=63^{\circ}\), \(m\overarc{JN}=180 - 63=117^{\circ}\). \(m\overarc{JLK}=360-(117 + 63)=180^{\circ}\)

Step3: Find \(m\overarc{JL}\)

Since \(\angle JPL\) is a right - angle (\(\angle JPL = 90^{\circ}\)), and the measure of an arc is equal to the measure of its central angle. So \(m\overarc{JL}=90^{\circ}\)

Answer:

1. \(63\)
2. \(180\)
3. \(90\)