Final answer:
Using the law of sines and the sum of angles in a triangle, we can find an unknown side of the triangle after determining the third angle, and subsequently calculate the area.
Step-by-step explanation:
To find the area of triangle ABC with AB = 12 cm, ∠A = 80°, and ∠B = 60°, we can use the formula for the area of a triangle which is ½ × base × height. However, since we only know two angles and one side, we first need to find the length of another side using the law of sines. Let side BC = a, AC = b, and AB = c. Then from the law of sines, we have:
a/sinA = b/sinB = c/sinC
To use this, we must first find the third angle ∠C using the fact that the sum of angles in a triangle is 180°. Since we know ∠A and ∠B, we can calculate ∠C = 180° - 80° - 60° = 40°.
Then we can apply the law of sines to find side AC (b):
b/sin(60°) = 12 cm/sin(40°)
Solving for b gives us the length of AC, which is the height of the triangle when AB is the base. With AB and the height, we can then calculate the area of the triangle.