-- First ...
You know that the angles of a triangle always add up to exactly 180° .
So you know that the two missing angles add up to (180° - 101.7°) = 78.3° .
That's the amount you need to split up in the ratio of 10:19 .
-- The smaller angle will get 10 shares of it.
The larger angle will get 19 shares of it.
Together, they'll add up to 29 shares of 78.3° .
-- OK. So what size is each share ? It's 78.3°/29 = exactly 2.7° .
-- The smaller angle gets 10 shares = (10 x 2.7°) = 27° .
The larger angle gets 19 shares = (19 x 2.7°) = 51.3° .
Check:
-- 27/51.3 = 10/19 <== good
-- (27° + 51.3° + 101.7°) = 180° <== good yay !