Thursday, December 5, 2024

Measuring Angles

I. Parts of a Plane Figure

A point in geometry is a location. It has no size i.e. no width, no length and no depth. A point is shown by a dot.

A line is defined as a line of points that extends infinitely in two directions. It has one dimension, length. Points that are on the same line are called collinear points.

A line is defined by two points and is written as shown below with an arrowhead.



A part of a line that has defined endpoints is called a line segment. Where a line extends infinitely in both directions a line segment is limited on both ends by the end points. 

A line segment can be written as:


A ray starts at one point and continues on forever in one direction.
                 
A ray can is written as:



II. Angle Introduction
An angle is formed when two straight lines or rays meet at a common endpoint. The common point of contact is called the vertex of an angle.





This figure is made up of two rays OA→ and OB→. The common end point of two rays is called the vertex of the angle. So, O is the vertex of angle AOB.
The rays OA→ and OB→ are called the arms or sides of angle AOB.
An angle is denoted by the symbol ∠. Only capital letters of English alphabets are used to name an angle. Name of angles can be written using three or one alphabet.
Thus, we can write the above angle in figure as ∠AOB or ∠BOA or ∠O.
We can see from the naming that vertex is always kept at the centre when written using three alphabets and only vertex when written as a single alphabet.

III. Angle Types

Right angles

A right angle is in the shape of a perfect corner, like the corner of a rectangular sheet of paper. Below is an example of a right angle. A right angle measures 90°
One ray extends to the right of the vertex. Another ray extends upward. A square at the vertex connects the rays to represent a perfect corner, or a right angle.

Straight angles

A straight angle looks like a straight line. Below is an example of a straight angle. A straight angle measures 180°
A straight line with a half circle on top and at the center of the line to represent a straight angle.

Acute angles

An acute angle is an angle whose degree measure is less than the right angle. Below is an example of an acute angle. An acute angle measures less than 90°
A ray extends to the right of the vertex. Another ray extends to the right and upwards from the vertex. A curved line connects the rays to represent an angle, in this case an acute angle, which has a degree measure less than 90 degrees.
When we compare an acute angle to a right angle, we can see that an acute angle is less than the right angle.
The acute angle from above set inside a 90 degree angle, called a right angle. Comparing the angles, the acute angle has a smaller degree of measure than the right angle.

Obtuse angles

An obtuse angle is an angle whose degree measure is greater than the right angle but less than the straight angle. Below is an example of an obtuse angle. An obtuse angle measures greater than 90° and less than 180°
A ray extends to the right of the vertex. Another ray extends up and to the left of the vertex. A curved line connects the rays to represent an angle, in this case an obtuse angle, which has a degree measure of greater than 90 degrees but less than 180 degrees.
When we compare an obtuse angle to a right angle, we can see that an obtuse angle is greater than 90°.
The obtuse angle from above with a 90 degree angle, called right angle, set inside it. Comparing the angles, the obtuse angle has a greater degree of measure than the right angle.
When we compare an obtuse angle to a straight angle, we can see that an obtuse angle is less than180°.
The obtuse angle from above set on a straight angle, or straight line. Comparing the angles, the obtuse angle has a smaller degree of measure than the straight angle.
Reflex angles
A reflex angle is an angle whose degree measure is greater than the straight angle but less than the complete angle. Below is an example of a reflex angle. A reflex angle measures greater than 180° but less than 360°
A ray extends to the right of the vertex. Another ray extends down and to the left of the vertex. A curved line connects the rays to represent an angle, in this case a reflex angle, which has a degree measure of greater than 180 degrees but less than 270 degrees.
When we compare a reflex angle to a straight angle, we can see that a reflex angle is greater than 180°.
The reflex angle from above set on a straight angle, or straight line. Comparing the angles, the reflex angle has a larger degree of measure than the straight angle.

Complete angles

A complete angle is a 360° angle. A complete angle has both rays pointing the exact same direction. That is because we've turned a full circle from the first ray. Below is an example of a complete angle.
A ray extends to the right of the vertex. Another ray also extends to the right of the vertex, exactly on top of the first ray. A curved line that forms a full circle connects the rays to represent an angle, in this case a complete angle, which has a degree measure of 360 degrees.

IV. Understanding Angle Measurement
There are two commonly used units of measurement of angles which are radians and degrees. In the case of practical geometry, we always measure the angle in degrees.

A degree is a unit of measure, denoted by the symbol °, used to indicate the measure an angle in a plane. An angle measuring 1°, read 1 degree, is equal to  of one complete revolution of the angle about its vertex. You can see from the diagram below that the counterclockwise rotation of the terminal side of an angle forms a circular path for the angle.

One full rotation is equivalent to 360°. One-quarter of a turn produces an angle that measures 90°, and one-half of a rotation creates an angle of 180°. While angles can have measures greater than what is shown, by doing multiple rotations, we will mostly only look at angle measures up to 360° in the study of Geometry.

Measure angles with a protractor

V. Measuring Angles 
Measuring angles using a protractor




Practice


Reference
https://mathsquery.com/geometry/fundamentals/angle-and-types/

https://www.splashlearn.com/math-vocabulary/geometry/angle

https://www.mathplanet.com/education/geometry/points-lines-planes-and-angles/measure-and-classify-an-angle

https://www.khanacademy.org/math/cc-fourth-grade-math/plane-figures/imp-angle-introduction/a/angle-types-review

https://www.math.net/degree
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