finding angle measures with only side lengths

You will have to use the Law of Cosines...and then the Law of Sines...calling the sides A, B  and C  and the angle beween sides A  and  B   =  θ  =  angle C

C^2  = A^2  + B^2  - 2(A)(B)cos [θ]       rearrange

cos [θ]  =  [ A^2 - B^2 - C^2] / [ 2(A)(B)]   = θ   

Now......use the cosine inverse to find  θ

arcos (  [ A^2 - B^2 - C^2] / [ 2(A)(B)]  )   = θ  = angle C

Now.....use the Law of Sines to find another angle.....

sin [angle C] /C  =  sin B / B       rearrange

sin B  =  B* sin[angle C] / C

And B  can be found using  the  sine inverse

arcsin [ B* sin[angle C] / C ]  = angle B

And angle A  = 180 - B -  C