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?

- - 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!