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This lecture covers the measurement of angles using a unit circle, defining angles in radians as the length of the arc cut by the angle, and the concept of trigonometric angles. It also discusses measuring angles with a protractor, examples of angle measurements in radians, and the movement of clock hands. The instructor explains the notion of trigonometric angles as the history of the rotation bringing the first branch onto the second. The lecture concludes with the description of circular motion, rotations around a point, and the infinite angles corresponding to these rotations.