Question

A triangle has an angle that measures 101.7°. The other two angles are in a ratio of 10:19. What are the measures of those two angles?

Answer 1

-- 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 !