Completing a shape using a line from existing shape - pen tool

Connect the two points at x to another point outside both shapes (i.e. right of A), select all, copy, paste in place, pathfinder→intersect.

Depending on what you want to do with these shapes, you could try this as one way:

Select the smaller shape and Object > Lock > Selection.

Select the larger shape and add anchor points where the smaller shape touches the lareger shape.

With the Direct Selection tool, select the anchor points on the part of the path you want to duplicate. Do not select the staert and end anchor points.

Copy the selected part of the path. Then choose Edit > Paste in Front.

Object > Unlock All.

Select the larger shape (careful not to select the copied path segment). And Object > Lock > Selection.

Select > All to select thew smaller shape and the path segment.

Object > Path > Join.

Object > Unlock All.

Now you will have two closed paths with matching edges.

Well, you may not consider this an easy way, but it will give you paths that are accurate.

The other shape should stay complete? In that case make a copy of that larger shape and then select both that copy and the new path and use the shapebuilder tool.

https://helpx.adobe.com/illustrator/using/creating-shapes-shape-builder-tool.html