Floor joists connection to eaves
After reading the question and all the comments, I have to add a few things. 

If in fact what you have are trusses not joists, then you need to know if they are load rated for a floor. Even if the space is only for storage, a rating of 60 to 80 pounds per sq ft is required. Storage can often become a heavier load that a person just walking on the floor.  Also if they are trusses, then the horizontal 2X4 or whatever is used is called a collar tie, not a joist. The key here is find the load bearing surface and size the new joists for the span. In this application with a limited support area will involve the height of the joist end supported. Often a ledger and post may be needed to bear the weight of the joist end. What is needed to really answer the question are accurate dimensions of vertical angle area above the supporting wall, span of the proposed floor etc.  The trusses in this question are probably designed to carry a roof load only, not a live load from a floor. Practically speaking, often sistered joists to collars will carry substantial loads, but the real question is how much and will they meet your local codes. A real concern is the fact that you have an existing ceiling, and any flex from weight above will effect it. 

My advice is to get your facts, measurements, and a professional opinion together before you do anything. Photos would help, but without doing the span math etc., no one can give you a viable solution from the info we have.