Question

find each measure.

1. ( m widehat{n j}=square )
2. ( m widehat{j l k}=square )
3. ( m widehat{j l}=square )
4. ( m widehat{j k}=square )

Explanation:

Step1: Calculate \(m\angle NPM\) and \(m\angle MPL\)

Since \(m\angle MPL = 45^{\circ}\) (given) and the circle has a total of \(360^{\circ}\), and \(\angle JPK = 90^{\circ}\).
Assume the four non - right - angle central angles are equal (symmetry of the figure). So \(m\angle NPM=m\angle MPL = 45^{\circ}\)

Step2: Calculate \(m\overset{\frown}{NJ}\)

The measure of an arc is equal to the measure of its central angle.
\(m\overset{\frown}{NJ}=m\angle NPJ\)
\(m\angle NPJ=m\angle NPM + m\angle MPL+\angle JPK\)
\(m\angle NPJ = 45^{\circ}+45^{\circ}+90^{\circ}=180^{\circ}\)

Step3: Calculate \(m\overset{\frown}{JLK}\)

\(m\overset{\frown}{JLK}\) is a major arc. \(m\overset{\frown}{JLK}=360^{\circ}-m\overset{\frown}{JN}\)
Since \(m\overset{\frown}{JN} = 180^{\circ}\), \(m\overset{\frown}{JLK}=360^{\circ}- 180^{\circ}=180^{\circ}\)

Step4: Calculate \(m\overset{\frown}{JL}\)

\(m\overset{\frown}{JL}=m\angle JPL\)
\(m\angle JPL=m\angle JPK + m\angle KPL\)
Since \(m\angle KPL = 45^{\circ}\) and \(m\angle JPK = 90^{\circ}\), \(m\angle JPL=90^{\circ}+45^{\circ}=135^{\circ}\)

Step5: Calculate \(m\overset{\frown}{JK}\)

The measure of an arc is equal to the measure of its central angle. \(m\overset{\frown}{JK}=m\angle JPK\)
Since \(\angle JPK = 90^{\circ}\), \(m\overset{\frown}{JK}=90^{\circ}\)

Answer:

1. \(180\)
2. \(180\)
3. \(135\)
4. \(90\)