Aright triangle (triangle 1) has a hypotenuse length of 72 mm and α = 52°. what are the measurements for a, b, and β?

Sounds as tho' there were one or more illustrations with this problem.  Could you possibly share the wealth?

If Triangle 1 has hyp 72 mm and one of the acute angles has measure alpha=52 deg., then the measure of the other acute angle must be (90-52), or 38 degrees.

We may now use the Law of Sines as a tool for calculating the lengths of the shorter sides alpha and beta:

        a            b                  c
= =
 sin alpha   sin beta      sin 90 deg.  

Here, 

c is the length of the hypo and is 72 mm; 

Therefore 

   72 mm            a
= applies to the given data.
 sin 90 deg       sin 52 deg

(72 mm)(sin 52) = a(sin 90) = 1, so

(72 mm)(sin 52) 
= a = 56.74 degrees
             1

and since the sum of the interior angles of the triangle must be 180, b = 180 - 56.74 deg., or beta = 33.26 deg.