diff --git a/src/mesh/mesh_triangle_holes.C b/src/mesh/mesh_triangle_holes.C
index 0250c1ea29d..9b7d5289930 100644
--- a/src/mesh/mesh_triangle_holes.C
+++ b/src/mesh/mesh_triangle_holes.C
@@ -123,7 +123,7 @@ namespace
     const Real denom = edgedx * raydy - edgedy * raydx;
 
     // divide-by-zero means the segments are parallel
-    if (denom == 0)
+    if (std::abs(denom) <= libMesh::TOLERANCE * libMesh::TOLERANCE)
       return -1;
 
     const Real one_over_denom = 1 / denom;
@@ -136,12 +136,13 @@ namespace
     const Real t = t_num * one_over_denom;
 
     // There's an intersection between the ray line and the edge?
-    if (t >= 0 && t < 1)
+    if (t >= -libMesh::TOLERANCE * libMesh::TOLERANCE &&
+      t - 1.0 < libMesh::TOLERANCE * libMesh::TOLERANCE)
       {
         // There's an intersection right on a vertex?  We'll count it
         // if and only if it isn't a "double-intersection", if the
         // *next* edge in line is on the other side of our ray.
-        if (!t)
+        if (std::abs(t) <= libMesh::TOLERANCE * libMesh::TOLERANCE)
           {
             const Real prevdx = edge_pt0(0)-ray_target(0),
                        prevdy = edge_pt0(1)-ray_target(1);