Find the measure of angle M.

Let's analyze the problem. We are given a circle with points P, M, and K on its circumference. Angle P is given as 84 degrees, and we need to find the measure of angle M.

Since PK is a chord of the circle, and angle P is an inscribed angle subtended by the arc MK, then angle M is also an inscribed angle subtended by the arc PK.

If PK passes through the center of the circle, then PK is a diameter, and angle M is a right angle. However, we don't know if PK passes through the center.

But we know that angle P is inscribed and it subtends the arc MK. Angle M is also inscribed and subtends the arc PK. Inscribed angles that subtend the same arc are equal. However, angle P subtends arc MK and angle M subtends arc PK. If PK is the diameter then angle M = 90 degrees. If PK is not the diameter, it can be any chord. But we know the sum of angles M and P should be 180 degrees.

Let's assume PK is a diameter of the circle. Then the angle MPK is inscribed in the circle and subtends the diameter. Thus, it is a right angle.

Then $$ \angle M + \angle P = 180^{\circ} $$.

Therefore, $$ \angle M = 180^{\circ} - \angle P = 180^{\circ} - 84^{\circ} = 96^{\circ} $$.

Answer: 96 degrees