The earlier mathematicians faced many problems in measuring the angle between two objects. To solve this problem, the famous mathematician Eudemus of Rhodes introduced the concept of the angle.
He measures the angle in a straight line. After that, the Carpus of Antioch measured the angle between intersecting lines. After that, Euild gives the concept of the angle. The angle names are often with \(\theta, \alpha, \beta, \gamma\) etc.
The word angle is derived from the Latin word “angulus,” which means the corner. The basic definition of the angle is the combination of two rays with a common endpoint is known as the angle.
Basic Concepts of Angles
Before we delve into the different types of angles, we should know the basic concepts. Angles are measured in degrees, the angle symbol is \((\circ)\). Radian is also a unit of angle, but we use degrees mostly. The meeting point of the two rays is called the vertex, while the rays themselves are termed arms.
Types of Angles
There are different types of angles are:
Acute Angles
The angle that is less than \(90^{\circ}\) is known as the acute angle.
Examples:
There are some acute angles, are:
$$45^{\circ},30^{\circ}, 75^{\circ}$$
Right Angles
The measure exactly at \(90^{\circ}\) is known as the right angle.
Obtuse Angles
The angle that is greater than \(90^{\circ}\) and less than \(180^{\circ}\) is known as an obtuse angle. In other words, an obtuse angle lies between \(90^{\circ}\) and \(180^{\circ}\).
The inequality form of the obtuse angle is:
$$90^{\circ} \lt \theta \lt 180^{\circ} $$
Examples:
Here are some examples given below:
$$95^{\circ},120^{\circ},165^{\circ}$$
Straight Angles
An angle that is equal to \(180^{\circ}\) is known as a straight angle. It is known as straight because it appears as a straight line.
Reflex Angles
The angle that is greater than \(180^{\circ}\) and less than \(360^{\circ}\) is known as a reflex angle. In other words, the reflex angle lies between \(180^{\circ}\) and \(360^{\circ}\).
The inequality form of the reflex angle is:
$$180^{\circ} \lt \theta \lt 360^{\circ} $$
Examples:
Here are some examples given below:
$$190^{\circ},220^{\circ},305^{\circ}$$
Complementary Angles
Two angles are complementary if their sum equals \(90^{\circ}\).
Mathematically Form:
$$\alpha+\beta=90^{\circ}$$
Example
30 degrees and 60 degrees are complementary because:
$$30^{\circ}+60^{\circ}=90^{\circ}$$
Supplementary Angles
On the other hand, supplementary angles add up to 180 degrees.
Mathematically Form:
$$\alpha+\beta=180^{\circ}$$
Example:
An example would be 120 degrees and 60 degrees.
$$120^{\circ}+60^{\circ}=180^{\circ}$$
Adjacent and Vertical Angles
Adjacent angles share a common vertex and side but do not overlap. Vertical angles are opposite each other and are always congruent.
Alternate Interior
In geometric figures with parallel lines, alternate interior angles lie on opposite sides of the transversal and are congruent.
Exterior Angles
In contrast, alternate exterior angles have the same properties but lie outside the parallel lines.
Coterminal Angles
Coterminal angles share the same terminal side in the standard position. They might have different measures but correspond to the same position on the coordinate plane.
what are Corresponding Angles?
The angles that occur on the same side of the transversal line and are equal in size is know as corresponding angles.
Converting Between Degrees and Radians
Understanding angle relationships is essential for advanced mathematical concepts. To convert from degrees to radians, multiply the degree measure by \(\frac{\pi}{180}\). It follows that the degrees to radians formula is:
$$\text{number of degree} \times \frac{\pi}{180}$$
The reverse involves multiplying radians by \(\frac{180}{\pi}\).
The radians to degree formula:
$$\text{number of radian} \times \frac{180}{\pi}$$
“Explore the fundamental concepts and principles of geometry in our article on What Is Geometry?“
Conclusion
In conclusion, angles are fundamental geometric entities formed by two rays with a common endpoint, known as the vertex. Derived from the Latin word “angulus,” meaning corner, angles play a crucial role in geometry and trigonometry. The concept evolved through contributions from mathematicians like Eudemus of Rhodes, Carpus of Antioch, and Euclid.
Angles are measured in degrees, with additional units like radians. The basic types of angles include acute, right, obtuse, straight, and reflex angles. Complementary angles sum up to 90 degrees, while supplementary angles add to 180 degrees.
Concepts like adjacent, vertical, alternate interior and exterior, coterminal, and corresponding angles further enrich our understanding of geometric relationships. Converting between degrees and radians facilitates advanced mathematical applications. Overall, the study of angles provides a foundation for diverse mathematical principles and applications in various fields.
“Understanding the various classifications of angles in geometry is fundamental to navigating spatial relationships and solving geometric problems.”
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