The diameter PК and two chords PN and PM equal to the radius of this circle are drawn through the point P of the circle.

The diameter PК and two chords PN and PM equal to the radius of this circle are drawn through the point P of the circle. Find the angles of the quadrilateral PMKN and the degree measures of the arcs PN NM MK PK.

Let’s draw a segment NO, the length of which is equal to the radius of the circle.

Since PN, by condition, is equal to R, then the PNO triangle is equilateral, therefore, all of its angles are equal to 60.

Similarly, the RMO triangle is also equilateral and all of its angles are 60, then the angle MPN = 60 + 60 = 120.

The inscribed angles PNK and PMK are based on the diameter of the circle, which means their values ​​are 90, then the angle NKM = 360 – 120 – 90 – 90 = 60.

The arc PN rests on the central angle PON, then its degree measure is 60.

The NM arc rests on the central angle NOM, then its degree measure is 120.

The MK arc rests on the inscribed angle of the MPC, the value of which is 60, then the degree measure of the arc is 120.

Arc PK is the diameter of a circle, then its degree measure will be equal to 180.

Answer: The angles of the PMKN quadrilateral are 120, 90, 60, 90, the degree measures of the arcs are PN = 60, NM = 120, MK = 120, PK = 180.