I have used Inkscape for this, but in a different manner to what SiliconWizard described.
You see, you must transform the image so that the measured item in the image is in the same plane as the image, i.e. perpendicular to the screen, with correct scaling on both axes. I import the image, and use Object > Transform until it is in a suitable orientation. I then add a new layer, and often use the Bézier curve tool to draw lines. I also use the File > Document Properties to set the page size so that I can easily measure things in millimeters in the image. (If you draw a line whose length you know, you can divide the document size by the relative error in the line length, to get the scaling exactly correct.)
The Measurement tool measures distances between two points, and the angle that line makes with the horizontal axis. You drag a "line" from the apex of the circle, to the non-horizontal end point of the arc.
When I want some arbitrary angle, I use the Circle tool. (If you press Shift+Control, you'll drag a "line" from the center of the circle to its perimeter.)
First, make sure View > Show/Hide > Tool Controls Bar is ticked. Then, draw a circle, either by pulling a diagonal of its axis-aligned bounding box, or by pressing Shift+Control and pulling a radius. You'll see the Tool Controls just below the menu bar. Make sure Rx and Ry are the same, so that you'll get a circular arc, not an elliptical one. Set Start: to 0, and End: to for example 45. Select the Select tool (simplest is to press ESC or F1), and click on it. The scaling arrows around it change into rotation arrows. Move the + in the center to the apex of the wedge; it is the rotation center.
You will want to use the Select tool and the rotation handles to adjust the initially upper edge of the wedge, and the arc end angle for the lower edge of the wedge. (Again, selecting the object, and then clicking on it with the Select tool, switches between rotation and scaling handles. Double-click to change to Tool Controls.)
The idea is to keep Start at zero, and only change the End angle, so that the End angle directly tells you the angle. It is clunky at first, but you quickly get used to it: just use the object rotation to set the counterclockwise edge, and the circle arc tools to set the clockwise edge of the arc. Whenever you have the Tool controls visible, you also see the radius of the arc; just keep Rx and Ry the same.
If you just use the arc control points –– which is easier and may be faster ––, you'll have to calculate the arc angle yourself, by substracting Start from End. They are always in degrees.
When you also have View > Show/Hide > Snap Controls Bar checked/visible, the "Snap an item's rotation center" (fifth up from bottom) is very useful, when you move the + in the center to the apex of the wedge, as it snaps precisely there then.