Enable Snap To Objects. (The red magnet icon on Infobar).

Using the Zoom Tool, (the magnifying glass) zoom in as close as you can to all the line segments.

Drag the top right line segment from the bottom end and snap it onto the top of the next arc. When the line ends are close to one another the cursor will change from a four-way arrow to a magnet to indicate a snap fit.

Continue working from the outside and moving clockwise towards the center snapping the next line segment to the previous.

You will be building a spiral. Be careful to snap each segment.

When you have a spiral like the one shown here, marquee select the entire group and from the Arrange menu, select Join.