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It depends on whether your connections are in the same terminal (most likely in B or C). If your connecting flight and arriving flight are both in B or both in C, then 35 min should be fine. If one flight is in C while the other flight is in B, then it might get a little tight. That underground tunnel between B & C can take a little time.
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I've made this kind of connection before, but I try to avoid a connection under 50 minutes at ORD, particularly in winter, unless i really need to. If everything is on time, you should be fine, even with a B to C connection (which I think is the norm for this kind of cnx, I believe most LGA flights leave/arrive into B, and LAX into C). I also try to make longer connections if I'm flying out of certain airports where on time % isn't great, LGA and EWR being two of them.
Of course, if for some reason you don't make the connection, UA will rebook you on the next available. The good news is that these two routes have frequent flights.
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This is just my opinion based on hundreds of flights in and out of ORD: you're asking for trouble with such a tight connection. ORD, just the UA part, is immense. Even if your flights are on time, you may have to walk (run?) from C4 to B21. If so, good luck! Look at the map of the airport online. United concourses are shown as Terminal 1, in Blue.
If your flight is late, you are totallz screwed. I mean totally. If it were me, I would not take the chance. The return is way too fight of a connection, especially if you have to go from B to C terminal. It is way too stressful. I would not do it, personally because of the stress during your whole flight out, once you are in the airport, and the torture that ensues if you miss that flight...
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Speaking as a definite non-expert with ORD (e.g. I've rarely walked by the Butch O'Hare display), I'd agree it depends where you're arriving and where you're departing. This recent thread about the bus shuttle between C and F offers some opinions on walking times as well.
Today flight 4 arrived about 26 minutes early into gate C9, and the flight to LGA took off a few minutes late from B2. Those two gates are about as far as you can get from one another in the United terminals.
Yesterday flight 4 arrived 2 minutes early at C 11 (after leaving over 20 minutes late). The flight to LGA left something like an hour and 20 minutes late from B7, again quite a distance. There's no way to even begin to guess what gates they will use, they constantly change.
If everything is on time, and if the gates are close to each other, you can make it. But anybody who has flown through Chicago can tell you you can land and spend a half hour or more driving around to get to the gates, driving right by your connecting flight pushing back (seriously, seen it happen). Can you make it? Yes, there's a chance. But there's lots that can go wrong. And there can be quite a distance between gates.
You can try to make it better, sign up for flight paging to your call phone so when you land in Chicago you will get a page telling you what gate you are arriving at and what gate your other flight is leaving from. Check out the airport maps so you have a guess where you are running off to, it will save you a few seconds at the monitors. Get a seat up front as much as you can, you can lose 15 minutes just trying to get off the plane.
The good thing is there are also flights at 2 and 3:05 in case you don't make it, if they have space they will put you on one of those. I would try to book the later flight to LGA to be safe