Handrail for stairs

[Robbox]

I want to install a new handrail to our stairs. The staircase does a 180 turn to reach the 1st floor landing, I think it's called a double winder, there is no 1/2 landing.
To fit the handrail so that it runs the full length of the main section, I'd like to fit the last couple of feet horizontally. I have worked out the angle I need to set on the cross-saw, but I'm not sure how to go about joining the pieces at that point.
I don't own anything like a dowel jig or biscuit jointer.
The handrail will be painted softwood, so I was thinking of using screws and glue.
I will fit handrail brackets either side of the join.

 

[Tigercubrider]

You can buy articulated brackets for this

But personally I'd buy a longish metal strap and bend it to the correct angle then screw it under the handrail
Most handrails come with a rebate and fillets to fit. Don't buy a strap that's too thick or you'll have to recess it. A shallow one is easier and you could sand the fillets to cover them

I would still use a dowel or peg and a fair amount of glue on the join. There should be enough wiggle room especially if you do trial fits

 

[Robbox]

Steel strap sounds a great way to strengthen the join.
Do you think I could make a careful drill hole on the top side from one piece to other and push a dowel through, after the strap is secured.
Glued it all up and sand down the dowel ends.

 

[Tigercubrider]

I'd use a dowel and drill carefully. You can probably enlarge one hole for corrections and add sawdust to PVA to hold the dowels.