Angle bisectors AX and BY of triangle ABC meet at point I. Find angle C, in degrees, if angle AIB = 106.

Since angle bisector AX divides angle A into two equal parts, then ∠AIB=∠BAI. Similarly, ∠AIB=∠BIC. Therefore, ∠BAI=∠BIC=21​⋅∠AIB=21​⋅106=53.

Since ∠BAI+∠BIC+∠BIC=180, then ∠BIC=180−53−53=74.

Since ∠C=∠BIC+∠BAI, then ∠C=74+53=127​.