Is there a simple strategy to determine whether a point is inside a boundary?

One way you can do this is to first discretize the graphics, then turn it into a region with an interior using DelaunayMesh, and finally using RegionMember:

mat1 = {{0., -5., 0}, {-5.2203, 0., 1.7945}}; 
mat2 = {{-0.8583, -4.9384, 0.1765}, {-5.4189, 0.7822, 2.3088}};
θ1 = 2.4798;
θ2 = 3.1275;
gr=
 Show[
  {ParametricPlot[
     mat1.{Sin[θ], Cos[θ], 1}, {θ, 0, 2.4798}, PlotStyle -> Red], 
   ParametricPlot[
     mat1.{Sin[θ], Cos[θ], 1}, {θ, 5.8629, 2 Pi}, PlotStyle -> Red],
   ParametricPlot[
     mat2.{Sin[θ], Cos[θ], 1}, {θ, 3.1275, 6.1325}],
   Graphics[
     {Line[{mat1.{Sin[θ1], Cos[θ1], 1}, mat2.{Sin[θ2], Cos[θ2], 1}}]}]}, 
  PlotRange -> All];

Now we turn the graphics into a region:

dg = DelaunayMesh@MeshCoordinates@DiscretizeGraphics@gr;

And we can now determine that {0,3} is inside this region:

RegionMember[dg, {0., 3.}]

True

Note that you can get better performance by creating a RegionMemberFunction and applying it directly to a list of points (it's Listable).

rf = RegionMember[dg]; (* create RegionMemberFunction *)
rf[{{0., 3.}, {0., 2.}, {5., 0.}, {0., 8.}, {1., 1.}}] (* apply the function *)

{True, True, False, False, True}

I am going to use Graphics`PolygonUtils`PointWindingNumber from R.M's answer to How to check if a 2D point is in a polygon?. In M10 you can use RegionMember.

The idea is to convert your region into a polygon and then test if the point is inside or not.

mat1 = {{0., -5., 0}, {-5.2203, 0., 1.7945}}; 
mat2 = {{-0.8583, -4.9384, 0.1765}, {-5.4189, 0.7822, 2.3088}};

q1 = 2.4798; q2 = 3.1275;
dq = 0.1; (*use smaller value for better precession*)
seg1 = Table[mat1.{Sin[q], Cos[q], 1}, {q, 0, 2.4798, dq}];
seg2 = Table[mat1.{Sin[q], Cos[q], 1}, {q, 5.8629, 2 Pi, dq}];
seg3 = Table[mat2.{Sin[q], Cos[q], 1}, {q, 3.1275, 6.1325, dq}];
seg4 = {mat1.{Sin[q1], Cos[q1], 1}, mat2.{Sin[q2], Cos[q2], 1}};
path = Join[seg1, seg2, seg3, seg4];

(*Make a continuous path from all the points*)
spath = FindCurvePath[path]//First; 
boundary = path[[#]] & /@ spath;

inPolyQ[poly_, pt_] := Graphics`PolygonUtils`PointWindingNumber[poly, pt] =!= 0


Manipulate[Graphics[Line[boundary], 
 PlotLabel -> Text[Style[StringForm["Point`` is ``", p, 
  If[inPolyQ[boundary, p], "Inside ", "Outside"]], Bold, Italic]]],
 {{p, {0, 0}}, Locator}]

For Mathematica9

For MMA9 you have to use Graphics`Mesh`PointWindingNumber[poly, pt] in place of Graphics`PolygonUtils`PointWindingNumber[poly, pt].

Perhaps a bit of an overkill. Imagine the points on the boundary and the points you input as electric charges:

mat1 = {{0., -5., 0}, {-5.2203, 0., 1.7945}}; 
mat2 = {{-0.8583, -4.9384, 0.1765}, {-5.4189, 0.7822, 2.3088}};
θ1 = 2.4798;
θ2 = 3.1275;
g=Show[
  {ParametricPlot[
     mat1.{Sin[θ], Cos[θ], 1}, {θ, 0, 2.4798}, PlotStyle -> Red], 
   ParametricPlot[
     mat1.{Sin[θ], Cos[θ], 1}, {θ, 5.8629, 2 Pi}, PlotStyle -> Red],
   ParametricPlot[
     mat2.{Sin[θ], Cos[θ], 1}, {θ, 3.1275, 6.1325}, PlotStyle -> Red],
   Graphics[
     {Thick, Red, Line[{mat1.{Sin[θ1], Cos[θ1], 1}, mat2.{Sin[θ2], Cos[θ2], 1}}]}]}, 
  PlotRange -> All];

step[form_] := 
 Function[# - 
    10^-4 Sum[
      Normalize[ext[[i]] - #]/Norm[ext[[i]] - #]^2, {i, 
       Length@ext}]] /@ form
relax[object_, t_] := First@{
   ext = RandomPoint[DiscretizeGraphics@g, 10^3];
   fin = NestList[step, object, t]; 
   Print@ListAnimate[
     Function@Show[g, #, Axes -> False] /@ 
      Function@ListPlot[#, PlotStyle -> Gray] /@ fin];
   Table[{Part[First@fin, i], 
      If[EuclideanDistance[Part[Last@fin, i], Mean@ext] - 
         EuclideanDistance[Part[First@fin, i], Mean@ext] < 0, 
       Text["Inside"], Text["Outside"]]}, {i, Length@First@fin}
     ] // MatrixForm
   }

Usage

Note that increasing t yields more accurate results.

relax[{{0, 0}, {2, 0}, {5, 6}}, 100]