Question 1137442
<br>
Draw a sketch with the 30 degree angle A at the left, one ray of the angle horizontal, and ray b upwards to the right at a 30 degree angle.<br>
{{{drawing(400,200,-2,12,-2,6,
line(0,0,11,0),line(0,0,5*sqrt(3),5),locate(-.5,-.5,A),locate(4,4,"b=10"),locate(5*sqrt(3),-.5,B),locate(5*sqrt(3),5.5,C),
line(5*sqrt(3),0,5*sqrt(3),5),locate(9,3,a)
)}}}<br>
If ABC is a right triangle, then a/b = sin(30) = 1/2, so a=5.<br>
If a is less than 5, no triangle is formed, because a is not long enough to reach from C to ray AB.<br>
If a is greater than 5, then two different triangles can be formed -- one with angle B acute and another with angle B obtuse.<br>
So the only value of a less than b for which there is a unique triangle with measurements A, a, and b is a=5.