The angles of a triangle are 1: 2: 3. Find the smaller side of this triangle if its larger side is 10 cm.

Let’s denote by x the smaller angle of this triangle.
According to the condition of the problem, the angles of the triangle are related as 1: 2: 3, therefore, the other two angles will be equal to 2 * x and 3 * x. Since the sum of the angles of any triangle is 180 degrees, the following relationship is true:
x + 2 * x + 3 * x = 180.
We solve the resulting equation:
6 * x = 180;
x = 180/6;
x = 30.
Knowing the smaller angle of this triangle, we find the rest:
2 * x = 60,
3 * x = 90.
Thus, this triangle is rectangular. In a right-angled triangle, the largest side is the hypotenuse, therefore, according to the condition of the problem, the hypotenuse of this triangle is 10 cm.
Let us denote by a the smaller side of this triangle. The smaller side of this triangle lies opposite the smallest angle, which is 30 degrees. Using the sine theorem, we can write:
a / sin (30 °) = 10 / sin (90 °).
Since sin (30 °) = 1/2, sin (90 °) = 1, we get:
a = 10 * 1/2 = 5.

Answer: The smaller side of this triangle is 5 cm.