## The definition of coterminal angles

A coterminal angle is an angle that shares the same terminal side as another angle. In other words, if you were to draw an angle next to another angle, and the two angles had the same endpoint, then those angles would be coterminal.

### The concept of coterminal angles

An angle is coterminal with another angle if the two angles share the same initial side and terminal side. Coterminal angles can be either positive or negative. The measurement of an angle is the number of degrees that it contains. The size of an angle is always positive, so coterminal angles will either both be positive or both be negative. A 135 angle is coterminal with a -225 angle, for example.

There are many ways to find coterminal angles. One way is to add or subtract 360 degrees from either angle until both angles are within 1 degree of each other; at that point, they are coterminal. Another way to find coterminal angles is to use the fact that any angle can be expressed as a multiple of 360 plus or minus some whole number. So, if one angle is 135 degrees, it can also be written as 135 + (360n) degrees, where n is any whole number. This means that any angle that is 135 + (360n) degrees is coterminal with a 135 degree angle; in other words, there are an infinite number of coterminal angles with a measure of 135 degrees.

### The measurement of coterminal angles

Coterminal angles are two angles that share the same initial side and terminal side. However, the size of the angles can be different. For example, 30° and 390° are coterminal angles because they share the same initial and terminal sides. The size of the angles is what differentiates them, with 30° being smaller than 390°.

Coterminal angles are important in mathematics because they allow for certain equations to be simplified. For instance, if you wanted to find the value of cos 75°, you could instead find the value of cos 15° because 15° and 75° are coterminal angles. This is helpful because cos 75° would be more difficult to calculate than cos 15°.

You can measure coterminal angles in degrees or radians. To measure an angle in degrees, you would use a protractor. To measure an angle in radians, you would use a calculator or a specific formula that involves pi (π).

Coterminal angles are all around us, so it’s important to understand how to identify and measure them!

## The angle 135 degrees

An angle is coterminal with another angle if the two angles have the same terminal side. The angle 135 degrees is coterminal with the angle 315 degrees, because both angles share the same terminal side. The angle 135 degrees is also coterminal with the angle -225 degrees.

### The definition of the angle 135 degrees

An angle is said to be coterminal with a given angle if it has the same initial and terminal sides. In other words, both angles have the same vertex and share a common side. Angles that are coterminal with each other differ in measure by a multiple of 360°. A 135° angle is coterminal with a -225° angle, a 405° angle, and any angle obtained by adding or subtracting 360° from these angles.

### The measurement of the angle 135 degrees

An angle is coterminal with another angle if the two angles have the same initial and terminal sides. In other words, if you were to draw the two angles on a piece of paper, they would have the same starting and ending points. Any two angles that share this property are coterminal with each other.

The measure of an angle is the measure of the interior of the angle. The most common unit of measure for angles is degrees, where a full circle is 360 degrees. Because angles that are coterminal with each other have the same initial and terminal sides, they also have the same measure. So, if one angle is coterminal with a 135 degree angle, then its measure must also be 135 degrees.

### The coterminal angles of the angle 135 degrees

There are an infinite number of coterminal angles for the given angle 135 degrees. Coterminal angles are defined as angles that share a terminal side. This means that the given angle 135 degrees and all other angles that are equal to or greater than 315 degrees and less than 486 degrees are coterminal with a 135 angle.

## The relationship between coterminal angles and the angle 135 degrees

When two angles have the same terminal side, they are coterminal angles. The angle 135 degrees is coterminal with the angles 225 degrees, -45 degrees, and 585 degrees. There are an infinite number of coterminal angles for the angle 135 degrees.

### The definition of the relationship

A coterminal angle is an angle that shares the same initial side and terminal side as another angle. In other words, they have the same angle “measure” (or size). Two angles can be coterminal even if they are measured in different units! As long as the angles have the same measure, they are coterminal.

For example, suppose we wanted to find an angle that is coterminal with a 135 degree angle. One way to do this would be to add or subtract 360 degrees until we get an angle between 0 and 360 degrees. So, 135 + 360 = 495 degrees is one possibility. OR, we could subtract 360 degrees twice: 135 – 360 = -225 degrees and -225 + 360 = 135 degrees, so -225 degrees is also coterminal with 135 degrees. Notice that 495 degrees and -225 degrees have the same measure (135 degrees), so they are both coterminal with 135 degrees.

Another way to find a coterminal angle would be to use the fact that any multiple of 360 degrees is also 360 degrees. So, if we take any integer and multiply it by 360, we will always get an angle that is coterminal with our original angle. For example, 2 * 360 = 720 so 720 degrees is coterminal with 135 degrees (because 720 = 495 + 225). We could keep going: 3 * 360 = 1080; 4 * 36 = 1440; etc. So 1080, 1440, 1800, 2160, 2520,… are all angles that are coterminal with 135 degrees.”

### The measurement of the relationship

Coterminal angles are two angles that share the same initial and terminal sides. There are an infinite number of coterminal angles for any given angle, and they can be found by adding or subtracting 360 degrees from the given angle. The most common coterminal angle is 360 degrees, which is also known as a full rotation. Other coterminal angles include: 180 degrees (a half rotation), 540 degrees (one and a half rotations), -180 degrees (- a half rotation), and -540 degrees (- one and a half rotations).

To find other coterminal angles, we can add or subtract 360 degrees as many times as we like until we find an angle between 0 and 360 degrees. For example, if we start with an angle of 30 degrees, we can add or subtract 360 degrees to get these results:

30 + 360 = 390°

30 – 360 = -330°

30 + 720 = 1050°

30 – 720 = -690°

We can continue this process until we find an angle between 0 and 360 degrees. In this case, it would be easier to subtract rather than add since the numbers are getting larger; eventually, we would get to an angle of 30 – 1080 = -1050°. The final answer would be -1050° since it is the only angle between 0 and 360 degrees.

### The importance of the relationship

Coterminal angles are two angles that have the same terminal side. The angle 135° is coterminal with the angle -45°, for example. Trust me, it’s important to know this stuff! In different branches of mathematics, you will encounter problems in which you need to find an angle that is coterminal with a given angle. In trigonometry, for instance, coterminal angles are used when working with the unit circle. There are an infinite number of coterminal angles for any given angle, so it’s often necessary to find the specific angle(s) that are needed for a particular problem. Here’s how it works:

There is a relationship between coterminal angles and the angle 135 degrees:

The angle 135 degrees is coterminal with the angle 45 degrees.

This means that if you were to start at the 0 degree mark on a circle and move clockwise 135 degrees, you would end up at the 45 degree mark. Likewise, if you started at the 0 degree mark and moved clockwise 45 degrees, you would also end up at the 45 degree mark. These two angles (135 degrees and 45 degrees) are said to be coterminal because they share the same terminal side (in this case, the side that falls on the 45 degree mark).