In a right-angled triangle ABC, the height CM and the bisector AK are drawn, intersecting at point O

In a right-angled triangle ABC, the height CM and the bisector AK are drawn, intersecting at point O. Find the angle B (in degrees) if the angle AOM is 77 °.

1. Let us introduce the designation of the angle by the symbol ∠.

2. The AMO triangle is rectangular, since the height of the CM forms a AMC equal to 90 ° with the side AB.

3. We calculate the value of ∠BAK, taking into account that the total value of all internal angles of the triangle is 180 °.

∠ВAK = 180 ° – ∠AMS – AOM = 180 ° – 77 ° – 90 ° = 13 °.

4. We calculate the value of ∠BAC, taking into account that the bisector AK divides it into two equal angles

∠ ВAK and ∠СAK:

∠ВАС = ∠ВАК x 2 = 13 ° x 2 = 26 °.

5. We calculate ∠АВС:

∠ABС = 180 ° – 26 ° – 90 ° = 64 °.

Answer: ∠В = 64 °.