solution: Since $\overline{OH}$ and $\overline{NH}$ are tangent to radii of the circle at $O$ and $N$, we have $\angle O =\angle N = 90^\circ$.  The sum of the measures of the interior angles of quadrilateral $JOHN$ is $360^\circ$, so $\angle J + \angle H = 360^\circ - \angle O - \angle N = \boxed{180^\circ}$.
prompt.content: <|im_start|>system
  You are a helpful assistant.<|im_end|>
  <|im_start|>user
  In circle $J$, $HO$ and $HN$ are tangent to the circle at $O$ and $N$. Find the number of degrees in the sum of $m\angle J$ and $m\angle H$. [asy]
  size(150);
  import graph;
  pair J = (0,0), H = (6,0), O, N;
  path circ = Circle(J,3);
  pair M = midpoint(J--H);
  path secCirc = Circle(M,3);
  pair[] tangentPoints = intersectionpoints(circ,secCirc);
  O = tangentPoints[0]; N = tangentPoints[1];
  draw(J--N--H--O--cycle);
  draw(circ);
  label("$H$",H,E);
  label("$J$",J,W);
  label("$N$",N,S);
  label("$O$",O,NE);
  [/asy]
  Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
  <|im_start|>assistant
  
prompt.role: user
extra_info.answer: 180^\circ
extra_info.question: In circle $J$, $HO$ and $HN$ are tangent to the circle at $O$ and $N$. Find the number of degrees in the sum of $m\angle J$ and $m\angle H$. [asy]
  size(150);
  import graph;
  pair J = (0,0), H = (6,0), O, N;
  path circ = Circle(J,3);
  pair M = midpoint(J--H);
  path secCirc = Circle(M,3);
  pair[] tangentPoints = intersectionpoints(circ,secCirc);
  O = tangentPoints[0]; N = tangentPoints[1];
  draw(J--N--H--O--cycle);
  draw(circ);
  label("$H$",H,E);
  label("$J$",J,W);
  label("$N$",N,S);
  label("$O$",O,NE);
  [/asy]
extra_info.split: test