solution: Since $\mathbf{a}, \mathbf{b},$ and $\mathbf{a-b}$ all lie in the same plane, from the diagram below, we see that the angle between $\mathbf{a}$ and $\mathbf{a} - \mathbf{b}$ is $84^\circ - 29^\circ = \boxed{55^\circ}.$
  
  [asy]
  unitsize(5 cm);
  
  pair A, B, C, O;
  
  O = (0,0);
  A = (1,0);
  B = extension(O, O + dir(29), A, A + dir(180 - 55));
  C = O + A - B;
  
  draw(O--A,red,Arrow(6));
  draw(O--B,green,Arrow(6));
  draw(C--A,green,Arrow(6));
  draw(O--C,blue,Arrow(6));
  draw(B--A,blue,Arrow(6));
  
  label("$\mathbf{a}$", A/2, S);
  label("$\mathbf{b}$", B/2, NW);
  label("$\mathbf{b}$", (A + C)/2, SE);
  label("$\mathbf{a} - \mathbf{b}$", C/2, SW);
  label("$\mathbf{a} - \mathbf{b}$", (A + B)/2, NE);
  label("$29^\circ$", (0.2,0.05));
  label("$55^\circ$", (0.15,-0.05));
  [/asy]
prompt.content: <|im_start|>system
  You are a helpful assistant.<|im_end|>
  <|im_start|>user
  Let $\mathbf{a}$ and $\mathbf{b}$ be vectors such that the angle between $\mathbf{a}$ and $\mathbf{b}$ is $29^\circ,$ and the angle between $\mathbf{b}$ and $\mathbf{a} - \mathbf{b}$ is $84^\circ.$  Find the angle between $\mathbf{a}$ and $\mathbf{a} - \mathbf{b}.$
  Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
  <|im_start|>assistant
  
prompt.role: user
extra_info.answer: 55^\circ
extra_info.question: Let $\mathbf{a}$ and $\mathbf{b}$ be vectors such that the angle between $\mathbf{a}$ and $\mathbf{b}$ is $29^\circ,$ and the angle between $\mathbf{b}$ and $\mathbf{a} - \mathbf{b}$ is $84^\circ.$  Find the angle between $\mathbf{a}$ and $\mathbf{a} - \mathbf{b}.$
extra_info.split: test