AM is the bisector of the right angle of an isosceles right-angled triangle ABC. Find the angles of the triangle ABM.

1) Since the ABC triangle is rectangular and isosceles, it means:
∠ А = 90 °;
∠ B = ∠ C = 45 °.
2) The bisector AM divides ∠ A in half, which means:
∠ BAM = ∠ MAC = 1/2 * 90 ° = 90 ° / 2 = 45 °.
2) Find the angles of the triangle ABM.
∠ А = ∠ BAM = 45 °;
The bisector AM of an isosceles right-angled triangle ABC to the base of BC is the height. The height is perpendicular to the base.
Hence, ∠ ВМА = 90 °.
We find ∠ МBА = 180 ° – 90 ° – 45 ° = 90 ° – 45 ° = 45 °.
Answer: triangle ABM: ∠ M = 90 °, ∠ A = ∠ B = 45 °.