Take the following example. Let's say we want all the blue lines to terminate at the red line.

First we need to select all the blue lines. Notice the dashed line around my selection window - this means only objects fully within this box are selected. This is achieved by holding down the Option key while using the Selection tool.

Next we choose the Extend Tool in the Edit bar. If, like me you prefer keyboard shortcuts to pushing buttons, then activate Extend by pressing +E.

Next, click on the red line...

And voilà!

Cheers.