ANGLESPPT
The Nature of Angles1.1. Definition of an AngleAn angle is formed by two ray...
The Nature of Angles1.1. Definition of an AngleAn angle is formed by two rays that originate from the same point and do not overlap. The point where the rays meet is called the vertex of the angle, and the extent of the angle is measured by the position of the rays relative to each other.1.2. Degree MeasurementThe standard way to measure the size of an angle is in degrees (°). One degree is equivalent to 1/360th of a full rotation, or 60 minutes. Angles can also be measured in radians ($\mathnormal{\rho}$), where one radian is equivalent to 57.3° or 1/2π full rotation.1.3. Types of AnglesRight anglesThese are angles that measure 90° and are denoted by a square at the vertexAcute anglesThese are angles that measure between 0° and 90°Obtuse anglesThese are angles that measure between 90° and 180°Straight anglesThese are angles that measure 180° and form a straight line when connectedComplementary anglesThese are angles that together form a right angleSupplementary anglesThese are angles that together form a straight angle1.4. Angular Measurement ToolsAngular measurement tools include protractors and compasses. A protractor is a device used to measure the angle between two lines or rays, while a compass is used to draw circles and measure the angle between two radii. Angles in Different Planes2.1. Angular Relationships in a Planar GeometryAngular relationships in planar geometry revolve around two fundamental relationships: congruence and similarity. Congruence means that two angles are the same size, while similarity means that two angles have the same shape but not necessarily the same size.2.2. Congruent AnglesCongruent angles are angles that have the same size and shape, but may not necessarily be the same angle. Congruent angles appear in many geometric shapes and configurations, such as triangles, rectangles, squares, and circles.2.3. Supplementary and Complementary AnglesTwo supplementary angles will form a straight angle, while two complementary angles will form a right angle. For example, if two angles each measure x degrees, and together they measure 90 degrees, then they are complementary angles. If two angles each measure y degrees, and together they measure 180 degrees, then they are supplementary angles.2.4. Central Angles and Polar AnglesA central angle is an angle formed by two radii meeting at the center of a circle, while a polar angle is an angle formed by two radii meeting at a point on the perimeter of a circle. Central angles and polar angles have important applications in geometry and trigonometry. Angular Measurement and Calculation Tools3.1. Protractors and CompassesA protractor is a tool used to measure the angle between two lines or rays, while a compass is used to draw circles and measure the angle between two radii. Protractors and compasses are fundamental tools for angular measurement and calculation in geometry and trigonometry.3.2. Angular Calculation Using Trigonometry FunctionsAngular measurement and calculation can also be performed using trigonometry functions such as sine, cosine, and tangent. For example, if you know the ratio of opposite over adjacent sides in a right triangle, you can use trigonometric functions to calculate the unknown measurements or solve for unknown angles.
