The bisectors of the angles A and B of the triangle ABC intersect at point M. Find the angle ACB if the angle AMB = 120 °

Considering the triangle ABC, we find:

<ACB = 180 ° – (<BAC + <ABC). Now consider the AMB triangle:

<AMB = 180 ° – (<BAM + <ABM) = 120 (<BAM + <ABM) =. (1)

Let’s define (<BAM + <ABM) = 180 ° – 120 ° = 60 °. (2)

Let us now consider the angles written in formula (1).

<BАМ = (1/2) * (<BАС); <ABM = (1/2) * (<ABC).

Summing up the obtained values, we get:

<BAM + <ABM = (1/2) * (<BAC) + (1/2) * (<ABC), whence we find from (2)

(<BAC + <ABC) / 2 = (<BAM + <ABM) = 60 °;

(<BAC + <ABC) = 2 * 60 ° = 120 °. Means,

<ACB = 180 ° – (<BAC + <ABC) = 180 ° – 120 ° = 60 °.

Answer: <ACB = 60 °.