Angles are geometric figures formed by two rays or line segments that share a common endpoint, known as the vertex of the angle. The word angles come from a Latin word called “angulus” meaning “corner.” Angles are used to measure the amount of rotation or separation between these two rays. They are usually measured in degrees, radians, or other angular units.

Definition of Angles:

Angles are geometric figures formed by two rays or line segments when meet at a common endpoint, known as the vertex of the angle. The word angles come from a Latin word called “angulus” meaning “corner.” Angles are used to measure the amount of rotation or separation between these two rays. They are usually measured in degrees, radians, or other angular units.

Symbol of Angles:

The symbol used to represent an angle in geometry is typically the Greek letter “θ” (theta) or sometimes “α” (alpha) or “β” (beta). The symbol is often placed at the vertex of the angle. For example:

• θ represents an angle as “theta.”

• α represents an angle as “alpha.”

• β represents an angle as “beta.”

These symbols are commonly used in mathematical equations, diagrams, and trigonometric functions to denote angles. The specific symbol used may vary depending on the context and personal preference.

Properties:

The angles have some common properties for understanding. The important properties are;

• The sum of the angles on one side of a straight line is always equal to 180 degrees.
• The sum of all the angles around a point is always equal to 360 degrees.

Types of Angles:

1. Acute Angle: An acute angle is an angle that measures less than 90 degrees.

2. Right Angle: A right angle measures exactly 90 degrees. It forms a square corner.

3. Obtuse Angle: An obtuse angle measures between 90 and 180 degrees.

4. Straight Angle: A straight angle measures 180 degrees, forming a straight line.

5. Reflex Angle: A reflex angle measures more than 180 degrees but less than 360 degrees.

6. Full Angle: A full angle, also known as a complete angle, measures 360 degrees. It represents one complete revolution.

7. Complementary Angles: Two angles are complementary if their measures add up to 90 degrees. For example, 30° and 60° are complementary angles.

8. Supplementary Angles: Two angles are supplementary if their measures add up to 180 degrees. For example, 120° and 60° are supplementary angles.

9. Vertical Angles: Vertical angles are formed when two lines intersect. They are opposite angles and are always congruent (have the same measure).

10. Adjacent Angles: Adjacent angles share a common vertex and a common side but do not overlap. The sum of adjacent angles is always 180 degrees.

11. Interior and Exterior Angles: Polygons typically use these terms. Interior angles are angles inside a polygon, while exterior angles are formed by extending one side of a polygon.

Parts of an Angle:

1. Vertex: The common endpoint where the arms of the angle meet.

2. Arms: The two rays or line segments that originate from the vertex and form the angle.

3. Interior of an Angle: The space enclosed by the two arms is considered the interior of the angle.

4. Exterior of an Angle: The space outside the angle but within the plane is the exterior of the angle.

5.Initial side: It is also known as reference line, because all the measurements are made taking this line as reference.

6.Terminal side: Terminal side is the side/ray up to which an angle measurement is done.

Examples:

• A right angle in a square measures 90 degrees.
• An acute angle, like in a triangle, can measure, for example, 30 degrees.
• A straight angle formed by a line is 180 degrees.
• A full angle representing a complete revolution is 360 degrees.
• Complementary angles, like 45° and 45°, add up to 90 degrees.
• Vertical angles in an X-shaped figure have equal measures.
• In a rectangle, adjacent angles are supplementary, adding up to 180 degrees.

Real-time Uses of Angles:

You can find the uses and applications of the angles almost everywhere. Angles have a wide range of real-life applications in various fields, including mathematics, science, engineering, and everyday activities.

 Some common uses are as such;

1. Architecture and Construction:

Architects and engineers use angles in the design and construction of buildings. Right angles and other specific angles are crucial for ensuring structural stability and aesthetics.

2. Navigation:

Nautical navigation relies on angles to determine the direction of travel and the position of ships. Concepts like latitude and longitude involve measuring angles.

3. Surveying and Landscaping:

Surveyors use angles to measure distances and map out land. Landscape designers use angles to plan the layout of gardens, walkways, and outdoor spaces.

4. Geography and Cartography:

Geographers and cartographers use angles to create maps and understand the Earth’s topography. Latitude and longitude lines are defined by angles with respect to the equator and prime meridian.

5. Astronomy:

Astronomers use angles to describe the positions of celestial objects in the sky. Concepts like right ascension and declination are used to locate stars, planets, and galaxies.

6. Mechanical Engineering:

Angles are critical in mechanical engineering for designing and analyzing mechanisms, such as gears, linkages, and levers.

7. Trigonometry:

Trigonometry, the study of angles and their relationships in right triangles is essential for solving problems related to heights, distances, and angles in various practical scenarios.

8. Navigation Systems and GPS:

GPS (Global Positioning System) devices use angles and satellite signals to determine a user’s precise location and calculate routes for travel.

9. Traffic Engineering:

Traffic engineers use angles to design road systems, including the curvature of roads, intersection angles, and traffic signal placement.

10. Photography and Cinematography:

Photographers and cinematographers use angles to frame shots, create visual effects, and control the composition of images and scenes.

11. Electrical Engineering:

Electrical engineers use angles to analyze alternating current (AC) waveforms, phasor diagrams, and electrical phase relationships.

12. Art and Design:

Artists and designers use angles to create perspective, depth, and balance in their compositions. The use of vanishing points and horizon lines involves understanding angles.

13. Sports:

Athletes, especially in sports like billiards, golf, and archery, use angles to calculate shot trajectories and improve accuracy.

14. Meteorology:

Meteorologists use angles to study the Earth’s climate and weather patterns. Concepts like sun angles affect the Earth’s temperature and weather conditions.

15. Carpentry and Woodworking:

Carpenters and woodworkers rely on angles for cutting, measuring, and joining materials. Angles are crucial for creating furniture, buildings, and other wooden structures.

16. Optics and Lenses:

Optics professionals use angles to understand the behavior of light as it passes through lenses, prisms, and mirrors, affecting the design of optical devices like cameras and eyeglasses.

Conclusion:

Understanding angles is essential for problem-solving, precision, and efficient design and planning in these and many other fields. In our lives without its applications, we may not perform anything.