In the first diagram, we are given that KM = 54 degrees and angle MDK = 62 degrees. Assuming that M and F are points on the circle, and D is a point outside the circle, with DM and DF being secant lines or tangents, and K is a point on the line segment DF. We are also given a red cross near point R. Without further information about the relationships between the points (e.g., if DF is a secant, or if DM is a tangent), it is impossible to determine specific values or prove geometric properties. However, if we assume that DM is a tangent to the circle at point M and DF is a secant intersecting the circle at K and F, and R is some other point on the circle, then angle MDK = 62 degrees is the angle formed by the tangent DM and the secant DF. The angle subtended by the arc MK at the circumference is not directly given. The value KM = 54 degrees is ambiguous. If it refers to an arc measure, then arc MK = 54 degrees. If it refers to an angle, it needs a vertex. Assuming it refers to arc MK = 54 degrees. Then the angle formed by the tangent DM and the chord MK would be half the angle subtended by the arc MK at the center, or equal to the angle subtended by the arc MK in the alternate segment. If angle MDK = 62 degrees, and it's the angle between a tangent and a secant, there's a theorem related to this, but we need more information about the secant's intersection points. The red cross at R is also unexplained. If we assume K coincides with F, and DM is a tangent at M, and DF is a secant passing through F, this contradicts the diagram. Let's consider the case where DM is a tangent at M and DF is a secant intersecting the circle at K and F. Then, angle MDK = 62 degrees. The angle formed by the tangent DM and the chord MK is not directly related to angle MDK in a simple way without more information. The notation KM = 54 degrees is unusual for an angle. If it refers to the arc MK, then arc MK = 54 degrees. The angle formed by the tangent DM and the chord MK would be half the measure of the arc MK, so angle DMK = 54/2 = 27 degrees. In triangle DMK, the sum of angles is 180 degrees. So, angle DKM = 180 - angle MDK - angle DMK = 180 - 62 - 27 = 91 degrees. However, angle DKM is an angle within the triangle formed by D, M, and K. The line DF is a secant. The diagram shows K is on DF. So, angle DMK is the angle between the tangent DM and the chord MK. The angle MDK = 62 degrees is given. If KM refers to arc MK = 54 degrees, then angle DMK = 27 degrees. In triangle DMK, angle DKM = 180 - 62 - 27 = 91 degrees. This seems plausible given the diagram. The red cross at R is likely an indicator for a point of interest or a part of a larger problem not shown. Without additional context or clarification on the notation KM = 54 degrees, a definitive solution is not possible. If KM refers to an angle formed by line segments from M to some point K, it's unclear. Assuming KM is arc measure. Let's reconsider. The notation suggests KM is an arc. If arc MK = 54 degrees, then the angle subtended by arc MK at any point on the circumference in the alternate segment is 54/2 = 27 degrees. The angle between the tangent DM and the chord MK (angle DMK) is equal to the angle subtended by the chord MK in the alternate segment. Thus, angle DMK = 27 degrees. In triangle DMK, the sum of angles is 180 degrees. Angle DKM = 180 - angle MDK - angle DMK = 180 - 62 - 27 = 91 degrees. This is a consistent interpretation. The red cross at R is extraneous information for this calculation.