7. In \( \triangle A B C \), bisectors of \( \angle \mathrm{A} \) and \( \angle \mathrm{B} \) intersect at point O . If \( \angle \mathrm{C}=70^{\circ} \). Find measure of \( \angle A O B \).

To find the measure of \( \angle A O B \), we can use the fact that the angle bisectors of a triangle intersect at a point called the incenter, and this point is equidistant from all sides of the triangle. The angle bisectors divide the triangle into six smaller angles, and the sum of these angles is equal to \( 360^{\circ} \). Let's denote the measure of \( \angle A \) as \( A \), the measure of \( \angle B \) as \( B \), and the measure of \( \angle C \) as \( C \). We are given that \( C = 70^{\circ} \). The sum of the angles in a triangle is \( 180^{\circ} \), so we have: \[ A + B + C = 180^{\circ} \] Substituting \( C = 70^{\circ} \) into the equation, we get: \[ A + B + 70^{\circ} = 180^{\circ} \] \[ A + B = 180^{\circ} - 70^{\circ} \] \[ A + B = 110^{\circ} \] Now, the angle bisectors of \( \angle A \) and \( \angle B \) divide these angles into two equal parts. Therefore, the measure of \( \angle A O B \) is the sum of the bisectors of \( \angle A \) and \( \angle B \), which is half of the sum of \( \angle A \) and \( \angle B \): \[ \angle A O B = \frac{A + B}{2} \] Substituting \( A + B = 110^{\circ} \) into the equation, we get: \[ \angle A O B = \frac{110^{\circ}}{2} \] \[ \angle A O B = 55^{\circ} \] So, the measure of \( \angle A O B \) is \( 55^{\circ} \).