Question

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

1. ( mwidehat{nj}=)
2. ( mwidehat{jlk}=)
3. ( mwidehat{jl}=)
4. ( mwidehat{jk}=)

Explanation:

Step1: Recall the property of vertical angles

Vertical angles are equal. Since \( \angle MPL\) and \( \angle NPJ\) are vertical angles, \( m\angle NPJ = m\angle MPL=63^{\circ}\). The measure of an arc is equal to the measure of its central angle. So \( m\overset{\frown}{NJ}=63^{\circ}\)

Step2: Find the measure of \( \overset{\frown}{JLK}\)

The total measure of a circle is \( 360^{\circ}\). The measure of a semicircle is \( 180^{\circ}\). \( \overset{\frown}{JLK}\) is a semicircle, so \( m\overset{\frown}{JLK} = 180^{\circ}\)

Step3: Find the measure of \( \overset{\frown}{JL}\)

We know that one of the central angles is a right - angle (\(90^{\circ}\)) and \(m\angle NPJ = 63^{\circ}\). \(m\overset{\frown}{JL}=m\angle JPL\). Since \(m\angle JPL=90^{\circ}+63^{\circ}=153^{\circ}\), so \(m\overset{\frown}{JL}=153^{\circ}\)

Step4: Find the measure of \( \overset{\frown}{JK}\)

We know that \(m\angle JPK\) and \(m\angle LPM\) are vertical angles. \(m\angle LPM = 90^{\circ}-63^{\circ}=27^{\circ}\). So \(m\overset{\frown}{JK}=27^{\circ}\)

Answer:

1. \(63\)
2. \(180\)
3. \(153\)
4. \(27\)