In △𝐴𝐵𝐶, 𝐴𝐵=6𝑐𝑚, 𝐴𝐶=12𝑐𝑚, 𝑚∠𝐴=60∘ and  𝐴𝐶― is the longest side. Which of the following statements about the measures of the angles in △𝐴𝐵𝐶 must be true? (Note: 𝑚∠𝑋 denotes the measure of angle X.)

Given that triangle ABC is a triangle with AB = 6cm, AC = 12cm, and ∠A = 60°, and AC is the longest side, we can infer that triangle ABC is a right triangle. This is because it satisfies the properties of a 30-60-90 triangle, where the side opposite the 30° angle is half the length of the hypotenuse, and the side opposite the 60° angle is √3/2 times the length of the hypotenuse.

1. Since ∠A = 60° and the sum of angles in a triangle is 180°, the sum of ∠B and ∠C is 120°.

2. Since AC is the longest side, ∠B must be the right angle (90°). This is because in a right triangle, the hypotenuse is opposite the right angle.

3. Therefore, ∠C must be 30° because the sum of angles in a triangle is 180°, and we already know that ∠A = 60° and ∠B = 90°.

So, the measures of the angles in triangle ABC are ∠A = 60°, ∠B = 90°, and ∠C = 30°.