An acute angle is an angle between 0 and 90 degrees (exclusive). Supplementary means that two angles add up to 180 degrees. So let’s see if we can find two acute angles whose sum is 180 degrees. The largest angles are 90+90=180. Anything less, for example, 89+89=178, add up to less than 180 degrees. But acute angles have to be less than 90, so even the largest angles cannot meet the requirements, therefore only one angle can be acute, the other being obtuse (more than 90 degrees and less than 180 degrees).
Two angles a and b are supplementary: a+b=180, so b=180-a. Angle b is acute: 0<b<90.
Angle a is acute: 0<a<90. Take a<90 then -90<-a. Add 180 to both sides: 180-90<180-a, 90<180-a. But b=180-a so 90<b, making b>90. But b<90 cannot be true. Therefore both a and b cannot be acute.