Final answer:
The sum of the remaining two angles in triangle KLM is 60 degrees. Because side KL is shorter than side LM, angle L is less than angle M, but without further information we cannot determine their exact measures.
Step-by-step explanation:
To find the sizes of the other two angles in triangle KLM, where angle K is 120 degrees and side KL is three-fourths the length of side LM, we first note that the sum of the angles in a triangle is always 180 degrees. Since one angle is already known to be 120 degrees, we subtract this from 180 degrees to find the sum of the remaining two angles:
180 degrees - 120 degrees = 60 degrees.
This sum is the combined total of angles L and M. Because the sides opposite these angles are in the ratio of 3:4 (since KL:LM is 3:4), we know that the angles opposite these sides are also in a ratio that follows the law of sines. However, since we do not have the measurements of the sides, we cannot apply the law of sines directly.
Instead, we assume that because side KL is shorter, angle L must be smaller than angle M. We let angle L be represented by x, which means angle M would be 60 - x. As we do not have exact lengths, we cannot determine a precise ratio to solve for x directly.
Therefore, without additional information such as side lengths or other angles, we cannot determine the exact measures of angles L and M, only that their sum is 60 degrees.