Solve it please---**Question:** 23. In the diagram below, sides OC and OB of triangle OBC are congruent. If the measure of angle OBC is 71 degrees, what is the measure in degrees of arc AD? **Diagram Description:** * **Type:** Geometric diagram involving a circle and a triangle. * **Shapes:** A circle centered at point O. A triangle OBC. * **Points:** Point O is the center of the circle and a vertex of the triangle. Points A and D are on the circle. Points C and B are vertices of the triangle, with B below the circle and C below the circle, to the left of B. * **Lines:** Line segment AO, line segment DO, line segment CO, line segment BO. Lines AD and CB intersect at O. * **Angles:** Angle OBC is labeled as 71 degrees. * **Labels:** Points are labeled A, D, O, C, B. The angle at B in triangle OBC is labeled 71°. * **Relative Position:** O is the center of the circle and also a vertex of the triangle. A and D are on the circle. C and B are vertices of the triangle below the circle. The lines AD and CB appear to be straight lines passing through O, implying they are diameters or chords. The diagram suggests that A, O, and some point on the line from C are collinear, and D, O, and some point on the line from B are collinear. Based on the question mentioning arc AD and triangle OBC, it is implied that AD and CB are chords/lines intersecting at O. The question also states OC and OB are sides of triangle OBC and are congruent. **Other Relevant Text:** * Sides OC and OB of triangle OBC are congruent. * The measure of angle OBC is 71 degrees. * The question asks for the measure in degrees of arc AD.

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Let's analyze this geometry problem step by step. We have a circle with center O, and triangle OBC where sides OC and OB are congruent. We're given that angle OBC measures 71 degrees, and we need to find the measure of arc AD. Let's analyze this geometry problem step by step. We have a circle with center O, and triangle OBC where sides OC and OB are congruent. Angle OBC measures 71 degrees. Points A and D are on the circle, and we need to find the measure of arc AD. Since sides OC and OB are congruent, triangle OBC is an isosceles triangle. In an isosceles triangle, the base angles are equal. Therefore, angle OBC equals angle OCB, both measuring 71 degrees. Using the angle sum property of triangles, we can find angle BOC: 180 minus 71 minus 71 equals 38 degrees. The key insight is understanding the positioning of points A and D. If A and D are positioned such that AO and DO are extensions of CO and BO respectively, then A, O, C are collinear and D, O, B are collinear. This means angle AOD equals angle COB, which is 38 degrees. Since the central angle equals the arc measure, arc AD measures 38 degrees. The key insight is that points A and D are positioned such that lines CB and AD intersect at point O. This creates vertical angles. Angle AOD and angle BOC are vertical angles, which means they are congruent. Therefore, angle AOD equals angle BOC, which is 38 degrees. Since the central angle equals the arc measure, arc AD measures 38 degrees. Let's summarize our solution. First, we identified that triangle OBC is isosceles since sides OC and OB are congruent. Second, we found that the base angles are equal, so angle OBC equals angle OCB, both measuring 71 degrees. Third, we calculated angle BOC using the triangle angle sum: 180 minus 71 minus 71 equals 38 degrees. Fourth, we applied the vertical angles theorem: angle AOD equals angle BOC, which is 38 degrees. Finally, since the central angle equals the arc measure, arc AD measures 38 degrees. Let's review the key geometric concepts we used to solve this problem. First, we applied isosceles triangle properties: when two sides are congruent, the base angles are equal. Second, we used the triangle angle sum theorem: the sum of angles in any triangle equals 180 degrees. Third, we applied the vertical angles theorem: vertical angles formed by intersecting lines are always congruent. Finally, we used the relationship between central angles and arcs: the central angle measure equals the arc measure. Therefore, our final answer is that arc AD measures 38 degrees.