Vertical Angles Exploration (2): Interactive Discovery of the Vertical Angles Th

① This interactive GeoGebra example dynamically demonstrates the fundamental geometric fact that vertical angles formed by two intersecting lines are equal. ② The core mathematical concept is the Vertical Angles Theorem, which states that the two pairs of opposite angles created by intersecting lines are congruent, along with angle measurement and comparison. ③ Construction highlights: using GeoGebra commands such as Line, Midpoint, Rotate, Mirror, Segment, Circle, Intersect, CircleSector, and Angle, the applet builds lines, midpoints, rotations, reflections, circles, intersection points, sectors, and angle objects. By dragging vertices or rotating lines, users can observe the real-time changes in angles α, β, γ, and δ to verify the equality of vertical angles. ④ This resource is ideal for middle school geometry lessons or self-study, serving as an inquiry-based exploration of the vertical angles theorem, and is also useful for teachers to demonstrate geometric properties through hands-on manipulation.