One side of the triangle is 1 m, and the angles adjacent to this side are 30 degrees and 45 degrees. Find the remaining sides.

Let side AB = 1m, Then <A = 30 degrees, <B = 45 degrees.
Let us denote the side AB = a, BC = b, AC = c.
Since the sum of the angles of the triangle is 180 degrees, we find the angle C, we get:
180- <B- <A = 180-45-30 = 105 degrees.
Find the sides we need through the area of the triangle:
S = 1/2 * a * b * sinA = 1/2 * a * c * sinB = 1/2 * b * c * sinC.
Let’s compose a system of equations:
b * sinA = c * sinB;
b * sinA = b * c * sinC;
From the second equation we find with:
sinC * c = sinA;
(√2 / 2) * c = 1/2;
c = √2 / 2.
Find b:
sin105 = (√6 + √2) / 4
1 / 2b = c * (√6 + √2) / 4;
b = (√3 + 1) / 2.
Answer: AB = 1m; BC = (√3 + 1) / 2m; AC = √2 / 2m.