用欧几里得几何方法分析解答这道题---```text 13. 如图, A, B, C 是⊙O上的三个点, ∠AOB=50° , ∠B=55° , 则∠A的度数为____. Chart/Diagram Description: * Type: Geometric figure (Circle). * Main Elements: * A circle with center labeled O. * Three points A, B, and C are on the circumference of the circle. * Line segments OA, OB, OC (radii) are drawn. * Line segments AB, BC, AC (chords) are drawn, forming triangle ABC. * The angle ∠AOB, formed by radii OA and OB at the center O, is indicated as 50°. * Points A, B, and C appear to be arranged in counterclockwise order on the circumference. Extraction Content: Question Number: 13 Question Stem: 如图, A, B, C 是⊙O上的三个点, ∠AOB=50° , ∠B=55° , 则∠A的度数为____. (As shown in the figure, A, B, C are three points on circle O, ∠AOB=50°, ∠B=55°, then the measure of ∠A is ____.) Interpretation of Question: * ⊙O denotes a circle with center O. * A, B, C are points on the circle. * ∠AOB = 50° is the central angle subtending arc AB. * ∠B=55° most likely refers to the inscribed angle ∠ABC = 55°. * ∠A likely refers to the inscribed angle ∠BAC. Calculation based on interpretation: * The inscribed angle subtended by arc AB is ∠ACB. ∠ACB = ∠AOB / 2 = 50° / 2 = 25°. * The inscribed angle ∠ABC = 55° is given. This angle subtends arc AC. The measure of arc AC is 2 * ∠ABC = 2 * 55° = 110°. * Assuming the points A, B, C are in order on the circle, the sum of the measures of arcs AB, BC, and CA is 360°. * Measure of arc AB = ∠AOB = 50°. * Measure of arc AC = 110°. * Measure of arc AB + Measure of arc BC + Measure of arc CA = 360°. * 50° + Measure of arc BC + 110° = 360°. * Measure of arc BC = 360° - 50° - 110° = 200°. * The inscribed angle ∠BAC subtends arc BC. Therefore, ∠BAC = Measure of arc BC / 2 = 200° / 2 = 100°. * Let's check the sum of angles in triangle ABC: ∠BAC + ∠ABC + ∠ACB = 100° + 55° + 25° = 180°. This is consistent. Answer (based on the calculation): 100 ```