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Angles are the measurements of degrees or radians from one line to another. Angles can be measured in degrees or radians, both of which attempt to capture the rotational distance in a circle.
Degrees are based on the idea that there are 3600 in a circle. The superscript "0" at the end of a number, from 00 to 3600 represents the word "degrees". Each of these 3600 is said to have 60' minutes, where the apostrophe denotes "minutes". And, each minute contains 60" seconds where the double hash mark denotes "seconds".
These marks are analagous to the use of such marks to denote "feet" with an apostrophe, as in [ 7' ] (seven feet) and in the symbol for inches ["], which denotes a subset of feet, as in [ 2" ] (two inches).
It should be noted that non-metric systems of measurement are much less effective and they make international engineering difficult.
Radians are based on the relationship between the radius of a circle and the corresponding arc length of the circle's perimeter.
In the sample circle to the right, the bright blue line represents the radius of a circle, which is the distance from the center point of a circle to a point on the perimeter of a circle.
The words "1 rad" that are just above the yellow curved line represent that the angle represented by the yellow line is 1 radian.
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When we have two lines that are separated by 1 radian, they isolate an arc on the perimeter of the circle, as in the bright magenta portion of the perimeter of the circle.
Given an angle of 1 radian, the radius of the circle and the arc length of the 1 rad arc are equivalent.
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In other words, the distance between the center of the circle and the edge of the circle is the same as the distance around the circle that is isolated by two radial lines separated by 1 radian of angle measurement.
One revolution that forms a circle, again, is 3600 or two times PI radians, or [2 rad].
is approximately 3.14, so 2 x is approximately 6.28 radians. There are, then, about 6.28 radian angles in an entire circle.
A line that is flat is said to have an angle of 1800, a line that is perfectly perpendicular to another line has an angle of 900.
A line that is evenly diagonal through a 900 angle is 450 from either of the two lines that make the 900 angle.
Angles that are 900 and are created by perpendicular lines are called right angles.
Angles that are less than 900 are called acute angles. And, angles that are greater than 900 but less than 1800 are called obtuse angles. |
A 1800 angle, or straight line, is called a straight angle. Any angle than is either acute or obtuse is referred to as an oblique angle. A zero angle is an angle in which two rays overlap and there is no angle between them
The above examples of angles are not displayed according to specific standards in geometry, however, as all angles are denoted by rays and by the specification of points at each vertex. The following examples show how angles and rays are supposed to be written with reference angles:
A vertex is the point at which an angle meets. It should be noted, also, that we can consider the outside angle, for instance, of the 450 acute angle above.
The exterior angle would simply be the measure of the remainder of the 3600 circle that is left after 450 has already been accounted for in the interior angle. This exterior angle is simply [3600 - 450], or 3150.
We can imagine that the angle in each of these figures to the right, which is represented by the light pink line, continues around as a whole circle.
There is an exterior angle that continues the interior angles. And, in cases of straight angles, both the intererior and exterior angles are 1800.
Knowledge of angles is important for understanding how functions work in more advanced mathematics such as trigonometry, which is the study of the angles related to the triangle. |
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