A circle with center 0 is inscribed into an angle with apex C equal to 68 °, which touches the sides

A circle with center 0 is inscribed into an angle with apex C equal to 68 °, which touches the sides of the angle at points A and B. Find the angle AOB.

Given: ∠C = 68 °; circle centered at point O; A, B – touch points.
Find: ∠AOB.
Decision.
Let’s draw a drawing for the task in accordance with the drawing in the photo.
1) Draw the radii OA and OB to the tangency points.
∠CAO = ∠CBO = 90 ° (by the radius theorem drawn to the tangency point).
2) ACBO is a quadrangle.
∠AOB = 360 ° – ∠ACB – ∠CAO – ∠CBO = 360 ° – 68 ° – 90 ° – 90 ° = 180 ° – 68 ° = 112 °.
Answer: ∠AOB = 112 °.