In △ABC, the measure of ∠A is 60°, the measure of ∠B is 45°, and the length of side AC is 8. What are the lengths of the other two sides? Do not use the Law of Sines! (You may use it afterwards to check.)
Although we could bang out the answers using the Law of Sines, we can find them using geometry as well. Draw altitude CD and observe that △ACD is a 30°-60°-90° right triangle and △CDB is an isosceles right triangle. Hence AD = 4 and CD = 4√3 → CB = (4√3)√2 = 4√6.
So CB = 4√6 and AB = 4 + 4√3.