In triangle ABC, AB = 12, BC = 6, angle A = 45, angle B = 90. Find side AC and angle C.

A triangle in which one angle is 90º is called rectangular.
In order to calculate the length of the AC side, we use the Pythagorean theorem, according to which the square of the hypotenuse is equal to the sum of the squares of the legs:
AC ^ 2 = AB ^ 2 + BC ^ 2;
AC ^ 2 = 12 ^ 2 + 6 ^ 2 = 144 – 36 = 108;
AC = √108 ≈ 10.4 cm.
Since the sum of all the angles of the triangle is 180º, then:
∠С = 180º – ∠А – ∠В;
∠С = 180º – 90º – 45º = 45º.
Answer: АС ≈ 10.4 cm; ∠С = 45º.