الإجابة النهائية:** الزوايا هي $\boxed{30^\circ}$, $\boxed{150^\circ}$, و $\boxed{270^\circ}$.

["The Final Answer: The Crucial Angles of 30°, 150°, and 270° in Geometry", "When solving geometric problems, identifying key angles can unlock the solution with clarity and confidence. Among the most significant angles encountered in triangles, trigonometry, and coordinate geometry are 30°, 150°, and 270°. This article explores why these angles are pivotal and how they serve as the final answer in numerous mathematical contexts.", "---", "### Why 30°, 150°, and 270° Matter in Geometry", "#### 1. The Angle of 30°: The Cornerstone of Equilateral Triangles and Trigonometry", "The angle 30° is foundational in geometry and trigonometry. Known as one-sixth of a full circle (360°), it frequently appears in:", "- Equilateral Triangles: Each internal angle measures 60°, so each angle at the base in a right triangle formed by bisecting an equilateral triangle is typically 30°.\n- 30-60-90 Triangles: This special right triangle features angles of 30°, 60°, and 90°—where side ratios rely fundamentally on 30° trigonometric functions like $\sin 30^\circ = \frac{1}{2}$ and $\cos 30^\circ = \frac{\sqrt{3}}{2}$.\n- Unit Circle: 30° corresponds to $\frac{\pi}{6}$ radians, essential for evaluating sine and cosine values used in calculus and engineering.", "Final Answer Insight: The 30° angle sets the baseline for many critical trigonometric identities and triangle solutions, making it indispensable.", "---", "#### 2. The Angle of 150°: Key to Reference Angles and Unit Circle Symmetry", "The 150° angle lies in the second quadrant, where sine is positive and cosine is negative. It represents:", "- The supplement of 30° ($180^\circ - 30^\circ$), so $\sin 150^\circ = \sin 30^\circ = \frac{1}{2}$, and $\cos 150^\circ = -\cos 30^\circ = -\frac{\sqrt{3}}{2}$.\n- A common angle in problems involving rotational symmetry, reflections, and phase shifts (e.g., in electrical engineering and physics).\n- A critical reference angle for calculating values across the unit circle.", "Final Answer Insight: 150° exemplifies how angles beyond basic triangle setups unlock deeper symmetry and periodicity in trigonometric functions.", "---", "#### 3. The Angle of 270°: A Pivotal Direction in Coordinate Systems", "Measuring 270° points straight downward along the negative y-axis in the standard position. This angle:", "- Defines a 270° rotation from the positive x-axis, crucial in polar coordinates and direction angles.\n- Corresponds to $\frac{3\pi}{2}$ radians, commonly used in parametric equations and circular motion.\n- Illustrates how angles define orientation and motion in geometry and physics.", "Final Answer Insight: 270° marks a perfect intersection of algebra, geometry, and coordinate systems, symbolizing a definitive directional answer.", "---", "### How These Angles Converge in Problem-Solving", "Whether analyzing triangle properties, evaluating trigonometric functions, or interpreting coordinate transformations, 30°, 150°, and 270° consistently emerge as fundamental angles. Together, they form a triad of pivotal values that:", "- Simplify calculations using symmetry and periodicity.\n- Enable students and professionals to apply trigonometry across science, engineering, and design.\n- Serve as the finally resolved angles that unlock exact solutions in both theoretical and applied mathematics.", "---", "### Conclusion: The Final Answer Is Clear", "In summation, the angles 30°, 150°, and 270° stand out as the final and most significant answers in geometric reasoning. Mastering these angles empowers deeper understanding and precise solutions in trigonometry and beyond. Remember:", "- 30° unlocks triangle ratios and trig basics\n- 150° reveals symmetry and reference angle strategies\n- 270° anchors rotational and directional calculations", "Embrace these angles, and unlock the confidence to tackle any geometric challenge with clarity.", "---", "Keywords: 30° angle, 150° angle, 270° angle, trigonometry, geometric angles, unit circle, 30-60-90 triangle, reference angles, coordinate geometry, final answer in math.", "Boxed Answer Summary:\nThe final angular solutions are:\n$$\n\boxed{30^\circ}, \quad \boxed{150^\circ}, \quad \boxed{270^\circ}\n$$\neach critical in geometry, trigonometry, and coordinate systems."]