In the triangle ABC AB = √2, BC = 2. Point M is marked on the side AC so that AM = 1, BM = 1. Find the angle ABC.

In triangle ABM, AB ^ 2 = (√2) ^ 2 = 2.

AM ^ 2 + BM ^ 2 = 1 + 1 = 2.

Then triangle ABM is rectangular, and MM is the height of triangle ABC.

Since the BM is height, the BCM triangle is also rectangular, in which the BM leg = 1 cm, and the BC hypotenuse = 2 cm, then the angle against the BM leg is 30.

Angle ACB = 30.

The ABM triangle is rectangular and isosceles, since AM = BM = 1 cm, then the angle MAB = MBA = 45.

Then in the triangle ABC the angle ABC = (180 – 45 – 30) = 105.

Answer: Angle ABC is 105.