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Geometry: Angles, Acute, Right, Obtuse, Complementary, Supplementary





 



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Angles

Angles are a fundamental component of geometry. They are used to determine the amount of difference in the directions of other geometric objects such as lines, rays, line segments and planes.

An angle is usually defined as follows:

Angle
An angle is the union of two rays that share the same endpoint.


An angle is the union of two rays at their endpoints.

Each ray is a side of the angle while the shared endpoint is called the vertex. There are many different types of angles that are classified in different ways. They may be classified by their relation to another angle or where they appear in a geometry figure. The most common way depends on how wide the rays are pointing away from each other.

Angle Measurements

Degrees

Angles are measured according to the divisions of the circle called degrees or radians.

A degree can be defined as follows:

Degree
A degree is 1/360 of a circle. It is 1/360 of the circumference and the area. It is measured from the center to the outer edge of the circle.


The circumference is the length of the outer edge of the circle while the area is inside.

A circle is marked off evenly in 360 divisions called degrees.

The degree is the most common unit of measurement for angles and is used to classify three fundamental types of angles. The symbol for a degree is °.

The acute angle is an angle whose degree measure is less than 90°.

An acute angle is between 0 an 90 degrees.

The right angle always has a measure of 90°. An object in geometry that is 90° to another is said to be perpendicular.

A right angle consists of perpendicular rays and is exactly 90 degrees.

An angle that is more than 90° is said to be an obtuse angle. Usually, this angle is between 90° and 180°.

An obtuse angle is between 90 and 180 degrees.

A 180° angle is sometimes called a straight angle which is basically just a straight line.

If an angle is greater that 180°, it may be called a reflex angle. It is used in special cases to count the number of degrees clockwise or counter-clockwise from a reference line.

Also, if an angle is 360° then it goes all the way around the circle back to 0°. So 0° and multiples of 360° are the same point on the circle.

Multiple Angles

Angles of Intersecting Lines

Another type of angle is called a vertical angle. When two lines intersect but in different directions, they create vertical and adjacent angles. The vertical angles are opposite each other while the adjacent angles are next to each other.

Vertical angles are a pair of angles that are opposite each other from the point of intersection of two lines.

Adjacent angles are a pair of angles that are next to each other with their vertex being the point of intersection of two lines.

Angles That Add To 90° and 180°

Complementary Angles
A complementary angle is an angle that adds to 90° when added to the measure of another angle.


Complementary angles are a pair of adjacent angles whose angle measures add to 90 degrees with their vertex being the point of intersection of three rays.

Supplementary Angle
A supplementary angle is an angle that adds to 180° when added to another angle.


Supplementary angles are a pair of adjacent angles whose angle measures add to 180 degrees with their vertex being the point of intersection of three rays.

The measure of an angle is indicated with the m and the angle symbol Angle Symbol together: mAngle Symbol.

When two lines intersect the adjacent angles are also supplementary angles.

The adjacent angles formed by the intersection of two lines are also supplementary angles.

The vertical angles have the same measure. Also, when two or more angles have the same measure they are called congruent. Therefore, it can be said that vertical angles are congruent.

In addition to being called supplementary angles, if the angles themselves share a side but still add to 180° then they are called a linear pair.

Bisectors

If an angle is split in two by a geometric object such as a line, ray or plane then the object is called the bisector if the two resulting angles are congruent. In other words, equal in measure.

Congruent angles are formed by the bisector of a larger angle. The bisector splits the larger angle into equal angle measures.

Bisectors also exist for other geometric objects such as line segments.

Notation

Point notation is often used to describe a particular angle.

Sometimes the letter representing the vertex is used such as Angle SymbolP but if there are three points in the angle with one at the vertex then all three points are used in the label such as Angle SymbolOPQ where the letter for the vertex would be in the middle.

Angle Postulates

Some of the postulates for angles include the following:

Protractor Postulate
There is only one positive real number for an angle between 0° and 180°.


Angle Addition Postulate
If two angles share a side then the sum is the measure of the angle of the other two sides.


Angle Bisector Theorem
A ray that bisects an angle divides its measure in half.


 




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