In triangle ABC, angle c with straight line AB = 6√3 cm, angle A = 60 degrees. Find BC and AC.

1. Angle C = 90 degrees, angle A = 60 degrees. By the theorem on the sum of the angles of a triangle, we find the degree measure of the angle В:
angle A + angle B + angle C = 180 degrees;
60 + angle B + 90 = 180;
angle B = 180 – 150;
angle B = 30 degrees.
2. In a right-angled triangle opposite an angle of 30 degrees, there is a leg, which is 2 times less than the hypotenuse. Since AB = 6√3 cm lies opposite the right angle C, then AB is the hypotenuse, then the leg AC (lies opposite the angle B) is equal to:
AC = AB / 2 = 6√3 / 2 = 3√3 (cm).
3. By the Pythagorean theorem:
BC = √ (AB ^ 2 – AC ^ 2) = √ ((6√3) ^ 2 – (3√3) ^ 2) = √ (36 * 3 – 9 * 3) = √ (108 – 27) = √81 = 9 (cm).
Answer: BC = 9 cm, AC = 3√3 cm.