Answer:
The measure of arc KD is;
[tex]51^{\circ}[/tex]Explanation:
Given the figure in the attached image.
Chord LD and MK intercept at N and also intercept the arc of the circle to form arc LM and KD.
[tex]\begin{gathered} \angle LM=209^{\circ} \\ \angle KD=24x+3 \end{gathered}[/tex]the angle LNM formed by the two chords is given as;
[tex]\angle LNM=66x-2[/tex]Recall that the angle formed by two intercepting chords can be calculated using the formula;
[tex]\begin{gathered} \text{ Angle formed by two intercepting chords = }\frac{1}{2}(\text{ sum of intercepted arc)} \\ \angle LNM=\frac{1}{2}(\angle LM+\angle KD) \end{gathered}[/tex]Substituting the given values;
[tex]66x-2=\frac{1}{2}(209+24x+3)[/tex]solving for x;
[tex]\begin{gathered} 66x-2=\frac{1}{2}(212+24x) \\ 66x-2=106+12x \\ 66x-12x=106+2 \\ 54x=108 \\ x=\frac{108}{54} \\ x=2 \end{gathered}[/tex]We have the value of x, let us now solve for the measure of arc KD by substituting the value of x;
[tex]\begin{gathered} m\angle KD=24x+3 \\ m\angle KD=24(2)+3 \\ m\angle KD=48+3 \\ m\angle KD=51^{\circ} \end{gathered}[/tex]Therefore, the measure of arc KD is;
[tex]51^{\circ}[/tex]