Exploring Angles in Paper Folding II

This GeoGebra applet explores changing angles in paper folding through geometric construction and symmetric transformations. Using polygons, angle bisectors, and mirror reflections, it dynamically demonstrates the angle relationships that arise during paper folding operations. The applet combines tangent functions and optimization problems to analyze the functional dependence between folding angles and related segment lengths, providing a rigorous mathematical framework for understanding these geometric phenomena. This interactive tool is designed for comprehensive instruction in geometric transformations and trigonometric functions at the middle and high school levels. It enables students to visually grasp the mathematical principles underlying paper folding, observing in real time how each fold generates precise angular and proportional relationships. By manipulating the folding parameters directly, learners can explore how changes in one angle propagate through the entire figure, reinforcing key concepts such as symmetry, congruence, angle bisectors, and trigonometric ratios. The integration of tangent-based analysis and extremum exploration further deepens students appreciation of how algebraic and geometric reasoning intersect. This applet is especially valuable for helping students connect abstract trigonometric identities to tangible, visual contexts, making complex topics like optimization and functional relationships more accessible and intuitive. It serves as an effective supplement to traditional geometry lessons on transformations and provides a hands-on platform for discovering the elegant mathematical structure embedded in everyday paper folding.