And Jesus returned (from being tempted by the devil) in the power of
the Spirit into Galilee: and there went out a fame of him through all
the region round about.
And he taught in their synagogues, being glorified of all. Luke
4:14-15
In Matthew we see the same timeline, Jesus being tempted
and then fame and then teaching: Matthew 4:1-24.
In Matthew we have a detailed account of what Jesus taught; Matthew 5 through 7.
Putting the accounts together we can understand
what Jesus meant by 'gospel' in Luke 4;
he was talking about his teachings.
The Spirit of the Lord is upon me because he has appointed me to
preach the gospel to the poor.... This days is this scripture
fulfilled in your ears Luke 4:18-21
After Jesus died and rose from the dead he tells his disciples to go into the world
and teach people his teachings.
Go ye therefore and teach all nations. Baptizing them... teaching them to observe
whatsoever I have commanded you. Matthew 28:19-20
The good news, the gospel, is what Jesus taught, his words, his teachings,
not just that he died and rose from the dead.
Believe on the Lord Jesus Christ and you shall be saved, and your
house.
And they told him all the word of the Lord and to all that were in his house.
... and was baptized, he and all his, straightaway. Acts 16:31-33
Notice that the apostles did just as Jesus had instructed in Matthew 28:19-20
they told them his teachings and they baptized them.
It is words such as 'give and it will be given to you...'
that are to be believed, Jesus himself is to be believed.
And therefore for the angel to have preached the gospel he would have needed to declare something about Jesus' teachings/way of life.