In mathematics, especially geometry, angles are formed by rays (or lines) that start at the same point or share the same endpoint. Angle measures the amount of rotation between the arms or the sides of an angle and is usually measured in degrees or radians. Where the rays intersect or meet is called the head.
Definition of angles.
Angles are the intersection or union of two rays at one point
In mathematics, an angle is the area bounded at the meeting point of two straight lines, and this region has a different size due to the different positions of the two lines with each other.
The angle is divided into five types: –
▪️ Sharp measures less than 90° .
▪️ A menu measuring 90°
▪️ The obtuse angle measures greater than 90° and less than 180°
▪️ °The straight line measures 180.
▪️ The reflex is greater than 180° and less than 360°.
Types of angles according to their relationships.
When the angles meet with each other, they work to form a certain angle, which is called certain names according to this intersection.
1. Integral angles.
▪️ This type of angle appears from its name that it works to complete each other to reach the straight angle of 180 degrees.
▪️ Usually, the integral angles are adjacent to each other and result in a straight line forming a straight angle, and both angles have the same side and vertex.
▪️ The two angles can be any type of angle. They can be both right, or one is sharp, the other is obtuse, and so on.
2. Complementary angles.
▪️ are the angles that also share the same side and vertex, but the difference here is that their sum is 90 degrees.
▪️ Complementary angles are similar to complementary ones, in that they must be next to each other, and this certainly explains why they share the rib and the vertex.
3. Adjacent angles.
▪️ Adjacent angles are the common denominator that combines both complementary and complementary angles, which is the confluence of the angles with the same vertex as well as the side.
▪️ This means that every two complementary angles are adjacent to each other, and it is not required that every two angles are adjacent complementary, and the same is the case for the complementary.
The type of angles in terms of direction.
The angle is determined by how it is measured clockwise or counterclockwise.
▪️ A positive angle is an angle that you can measure counterclockwise through the positive x-axis.
▪️ Negative angle, the exact opposite of a positive angle, is measured with the same clockwise direction and is equal to the same positive angle value but with the opposite sign.
Relationship between angles.
There are 8 relationships between angles: –
1) Complementary angles.
If the sum of the two angles is 90°, they are called complementary angles.
2) The two complementary angles.
The sum of the two complementary angles = 180°.
3) The two adjacent angles.
In order to call the two angles adjacent to each other, they must fulfill the following conditions.
A) That they have a common head.
B) to have a common rib.
C) That the two angles are on both sides of the common side.
4) The opposite corners of the head.
The two opposite angles of the vertex are equal.
Conditions for opposite angle to head.
A) To be shared in the same head.
B) Their sides are on the same length.
5) The two alternating angles.
They are all the two angles that lie on two sides with the cutter and fall within two other lines and form approximately the letter Z, the two alternating angles are equal.
6) The two corresponding angles.
They are all two angles that lie on the same side of the cutter and one of them lies inside two lines and the other is outside it, forming the letter F approximately, and they are equal.
7) The two allied angles.
They are all two angles located on the same side of the cutter, and they are both inside the other two lines and form a letter U roughly and they complement each other, the sum of its measure equals 180°.
8) The outer corner of a triangle.
Is the angle whose first side is the side of the triangle and the other side is an extension of the other side.
Sum of the angles.
When we have two angles, for example 25 ° and 65 °, we can calculate their sum by simply adding them as follows:
The sum of the two angles is 90∘ = 65∘ + 25∘ =In this example the sum of the two angles together is 90 °, which means that they form a right angle.
The ability to calculate the sum of angles is important when studying the properties of various geometric shapes, such as squares and triangles.
The angle represents a mathematical quantity, and therefore it must have a unit of measurement that distinguishes it, but it is distinguished from other mathematical quantities by having more than one unit of measurement, which can be dealt with as follows:
Assuming that the line connecting the two sides of the angle has taken a curved shape, i.e. a sector in a circle, the minimum length of this line is zero, and the maximum length of this sector is the circumference of a circle, and this perimeter has been divided into 360 parts, each part is called a “degree” And each degree was divided into 60 parts, each part was termed “minutes”, and every minute was divided into 60 parts, each part was called the term “second”, and the degree was the most famous unit of measure of the angle, and it was called the sexagesimal system, in order to divide Class to 60 minutes, 60 seconds.
It is the second unit of angleIt is called the circular system, and the radians are the central angle in a circle, the length of the arc facing it is equal to the radius of that circle, meaning that the circular angle is equal to 2 radians, and the right angle in it is equal to half a radian.
In this system, the circular angle is divided into 400 parts, and each part is called a term “grad” or “locust”, and the right angle in it is equal to 100 grad.Thus, we see that the angle has several measurement systems, the units of which depend on the type of system used in the measurement process.
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