In right ∆abc, m b = 90°, m c = 40°, and bc = 10. what are the other two side lengths of the triangle?

Angle at point A = 180 - 90 - 40 = 50 degrees
To find 2 remaining lengths of sides of triangle we just need to use basic trigonometry.

cos40 = 10/AC   =>  AC = 10/cos40 = 13.05
tan40 = AB/10   =>   AB =  10*tan40 = 8.39

If we want to check this we can use Pythagoras theorem:
AB^2 + BC^2 = AC^2   and see that lengths that we calculated are correct.