Measurements of Angles: Angles are figures formed by two rays, which are the angles’ sides and share a common initial point, known as the angle’s vertex. The angles formed by two rays are located in the plane containing the rays. Angle measurement is the amount of rotation the ray makes from its starting point to its ending point. Angles can be measured using a wide range of units. If the rotation is clockwise, the angle is positive; if the rotation is anticlockwise, the angle is negative. 

This article will study the definition of angles, types of angles, and some angle relations and solve some example problems. Read on to find out more.

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Angles

An angle is the union of two non-collinear rays with a common initial point.
The two rays forming an angle are called the arms of the angle, and the common initial point is called the vertex of the angle.
The angle formed by the rays \(AB\) and \(AC\) as shown below is denoted by \(\angle BAC\) or \(\angle CAB.\)
In the above figure, \(\angle BAC\) divides the plane containing it into two parts:
(i) The part of the plane which lies between the rays \(AB\) and \(AC\)
(ii) The part of the plane which does not lie between the rays \(AB\) and \(AC\)
The first part is known as the interior of \(\angle BAC,\) and the second part is the exterior of \(\angle BAC.\)
For the sake of convenience, the angle \(\angle BAC\) or \(\angle CAB\) can also be referred simply as \(\angle A.\) However, this cannot be done if there is more than one angle with the same vertex \(A.\) In such cases, we can denote the angles by the symbols such as \(\angle 1,\angle 2,\angle 3,\) etc., as shown below.

Interior of an angle: The interior of an angle \(BAC\) is the set of all points in its plane.

The exterior of an angle: The exterior of an angle \(BAC\) is the set of all points in its plane, which do not lie on the angle or in its interior.
Congruent Angles: Two angles are congruent if a traced copy of one can be superposed on the other to cover it completely and exactly.

Example: If \(\angle BAC\) is congruent to \(\angle FEG,\) then we write \(\angle BAC \cong \angle FEG.\) Congruent angles will be equal angles, and we shall write \(\angle BAC = \angle FEG\) instead of writing \(\angle BAC \cong \angle FEG.\)

Types of Angles

This section will define various angles based on their measure and the relation of their measures with the measure of some given angle.

Right angle: An angle whose measure is \({90^ \circ }\) is called a right angle.

Acute angle: An angle whose measure is less than \({90^ \circ }\) is called an acute angle. Thus, an angle whose measure is less than a right angle is an acute angle.

Therefore, \(\angle BAC\) is an acute angle if \(0 < \angle BAC < {90^ \circ }\)

Obtuse angle: An angle that measures more than \({90^ \circ }\) but less than \({180^ \circ }\)is called an obtuse angle. Thus, an angle whose measure is more than a right angle is an obtuse angle.

Therefore, \(\angle BAC\) is an obtuse angle, if \({90^ \circ } < \angle BAC < {180^ \circ }\)

Straight angle: An angle whose measure is \({180^ \circ }\) is called a straight angle.

Reflex angle: An angle whose measure is more than \({180^ \circ }\) is called a reflex angle. The measure of a reflex angle is more than \({180^ \circ }\) and less than \({360^ \circ }.\)

Complementary angles: Two angles, the sum of whose measures are \({90^ \circ },\) are called complementary angles.
Thus, \(\angle BAX\) and \(\angle XAC\) are complementary angles, if \(x + y = {90^ \circ }\)

Supplementary angles: Two angles, the sum of whose measures is \({180^ \circ },\) are called supplementary angles.

Angle Bisector: A ray \(AX\) is said to be the bisector of \(\angle BAC,\) if \(X\) is a point in the interior of \(\angle BAC,\) and \(\angle BAX = \angle XAC\)

Some Angle Relations

In this section, we will study the relations between the angles of a figure.

Adjacent angles: Two adjacent angles; if they have the same vertex, they have a common arm, and uncommon arms are on either side of the common arm.

In the above figure, \(\angle AOC\) and \(\angle BOC\) have a common vertex \(O.\) Also, they have a common arm \(OC\) and the distinct arms \(OA\) and \(OB,\) lie on the opposite sides of the line \(OC.\) Therefore, \(\angle AOC\) and \(\angle BOC\) are adjacent angles.

Linear pair of angles: Two adjacent angles form a linear pair of angles if their non-common arms are two opposite angles.

In the below figure, \(OA\) and \(OB\) are two opposite rays, and \(\angle AOC,\angle BOC\) are the adjacent angles.
Therefore, \(\angle AOC\) and \(\angle BOC\) form a linear pair.

Measure of an Angle

We use a graduated ruler or tape to measure the length of a line segment. Similarly, we use a protractor to find the measure of an angle. In this section, we shall list some important facts about the measure of an angle.

Angle measure axiom: Every angle has a measure. The unit of angle measure is called a degree. The measure of an angle in degrees is a real number lying between \(0\) and \(360.\)
If the measure of \(\angle BAC\) is \(x\) degrees, we denote it be \(\angle BAC = {x^ \circ }\)

Congruent angle measure axiom: Two congruent angles have the same measure, and conversely, two angles of equal measure are congruent.

Angle Addition axiom: If \(X\) is a point in the interior of \(\angle BAC,\) then \(\angle BAC = \angle BAX + \angle XAC\)

Angle construction axiom: Given a ray \(AB\) in a plane, and a real number \(x,\) lying between \({0^ \circ }\) and \({180^ \circ },\) there exist two rays, \(A{C_1}\) and \(A{C_2},\) with \({C_1}\) and \({C_2}\) lying on the opposite side of \(AB\) such that \(\angle BA{C_1} = \angle BA{C_2} = \angle BAC.\) In other words, given a ray \(AB,\) we can always construct an angle of the given measure on either side of \(AB.\)

Different Systems of Measurement of Angles

There are three units of measure for angles: revolutions, degrees, and radians. In trigonometry, radians are used most frequently, but it’s important to convert between any three units.

Measurement of Angles in Trigonometry

The following three systems of units are used in the measurement of trigonometrical angles:

(i) Sexagesimal System (or English System)
(ii) Centesimal System (or French System)
(iii) Circular System

Sexagesimal System: In this system, an angle is measured in degrees, minutes, and seconds.

Centesimal System: In this system, an angle is measured in grades, minutes, and seconds. In this system, a right angle is divided into \(100\)

Circular System: In a circular system, an angle is measured in radians. In higher mathematics, angles are usually measured in a circular system. In this system, a radian is considered as the unit for the measurement of angles.

Angle Measurement Units

The revolution is that the most natural unit of measurement for an angle. It’s defined because of the amount of rotation required to travel from the initial side of the angle all the way around back to the initial side. A method to see a revolution is to imagine spinning a wheel around just one occasion.

The space travelled by any point on the wheel is adequate to one revolution. An angle can then tend a worth supported the fraction of the space some extent travels divided by the space travelled in one rotation. For instance, an angle represented by \(\frac{1}{4}\) turn of the wheel is adequate to \(0.25\) rotations.

A more common unit of measurement for an angle is that the degree. This unit was employed by the Babylonians as early as \(1,000\) B.C. At that point, they used a variety system that supported the amount \(60\), so it had been natural for mathematicians of the day to divide the angles of an equiangular triangle into \(60\) individual units. These units became referred to as degrees.

The notation for an angle of \(32\) degrees, \(19\) minutes, and \(30\) seconds is \(32^\circ ,19’30”\).

Another unit of angle measurement used extensively in trigonometry is the radian. This unit relates a singular angle to every real. Consider a circle with its centre at the origin of a graph and its radius along the \(x\)-axis. One radian is defined because the angle created by a levorotation of the radius around the circle such the length of the arc travelled is adequate to the length of the radius.

Solved Examples on Measurements of Angles

Q.1. In the below figure, OA and OB are opposite rays:
(i) If \(x = 75,\) what is the value of \(y\)?
(ii) If \(y = 110,\) what is the value of \(x\)?

Ans: Since \(\angle AOC\) and \(\angle BOC\) form a linear pair.
\(\angle AOC + \angle BOC = 180^\circ \,\,\,\,\,…\left( {\rm{i}} \right)\)
(i) If \(x = {75^ \circ },\) the from \(\left({\text{i}} \right)\) \(75 + y = {180^ \circ } \Rightarrow y ={105^ \circ }\)
(ii) If \(y = {110^ \circ },\) then from \(\left({\text{ii}} \right)\)\(x + {110^ \circ } = {180^ \circ } \Rightarrow x ={70^ \circ }\)

Q.2. In the below figure, \(\angle AOC\) and \(\angle BOC\) form a linear pair. Determine the value of \(x\)

Ans: Since \(\angle AOC\) and \(\angle BOC\) form a linear pair.
Therefore, \(\angle AOC + \angle BOC = {180^ \circ }\)
\( \Rightarrow 4x + 2x = {180^ \circ }\)
\( \Rightarrow 6x = {180^ \circ }\)
\( \Rightarrow x = \frac{{{{180}^ \circ }}}{6} = {30^ \circ }\)

Q.3. In the below figure, OP bisects \(\angle BOC\) and \(OQ,\angle AOC.\) Show that \(\angle POQ = {90^ \circ }\)

Ans: Since \(OP\) bisects \(\angle BOC\)
Therefore, \(\angle BOC = 2\angle POC\)
Again, \(OQ\) bisects \(\angle AOC\)
Therefore, \(\angle AOC = 2\angle QOC\)
Since ray \(OC\) stands on line \(AB\)
Therefore, \(\angle AOC + \angle BOC = {180^ \circ }\)
\( \Rightarrow 2\angle QOC + 2\angle POC = {180^ \circ }\)
\( \Rightarrow 2\left({\angle QOC + \angle POC} \right) = {180^ \circ }\)
\(\Rightarrow \angle QOC + \angle POC = {90^ \circ }\)
\( \Rightarrow \angle POQ = {90^ \circ }\)

Q.4. In the below figure, lines \({l_1}\) and \({l_2}\) intersect at \(O,\) forming angles as shown in the figure. If \(a = {35^ \circ },\) find the values of b,c and d

Ans: Since lines \({l_1}\) and \({l_2}\) intersect at \(O\)
Therefore, \(\angle a = \angle c\) (Vertically opposite angles)
\(\angle c = {35^ \circ }\)
Clearly, \(\angle a + \angle b = {180^ \circ }\)
\( \Rightarrow {35^ \circ } + \angle b = {180^ \circ }\)
\( \Rightarrow \angle b = {180^ \circ } – {35^ \circ }\)
\( \Rightarrow \angle b = {145^ \circ }\)
Since \(\angle b\) and \(\angle d\) are vertically opposite angles.
Therefore, \(\angle d = \angle b\)
\(\Rightarrow \angle d = {145^ \circ }\)
Therefore, \( \Rightarrow \angle b = {145^ \circ },\angle c = {35^ \circ }\) and \(\angle d = {145^ \circ }\)

Q.5. In the below figure, OA,OB are opposite rays and \(\angle AOC + \angle BOD = {90^ \circ }.\) Find \(\angle COD.\)

Ans: Since \(OA\) and \(O\) are opposite rays. Therefore, \(AB\) is a line since ray \(OC\) stands on line \(AB.\)
Therefore, \(\angle AOC + \angle COB = {180^ \circ }\)
\( \Rightarrow \angle AOC + \angle COD + \angle BOD = {180^ \circ }\,\left({\angle COB = \angle COD + \angle BOD} \right)\)
\( \Rightarrow \,\left({\angle AOC + \angle BOD} \right) + \angle COD = {180^ \circ }\)
\( \Rightarrow \,{90^ \circ } + \angle COD = {180^ \circ }\)
\( \Rightarrow \,\angle COD = {90^ \circ }\)

Summary

This article taught us the definition of angles, different angles, and relations between angles like adjacent angles and linear pairs of angles. Also, we have learned the types of measurement of angles and their axioms like angle measure axiom, congruent angle measure axiom, angle addition axiom, and angle construction axiom and solved some example problems based on the same.

Learn Different Types of Angles

FAQs on Measurements of Angles

Q.1. How do you find a measurement of an angle?
Ans: We use a graduated ruler or tape to measure the length of a line segment. Similarly, we use a protractor to find the measure of an angle.

Q.2. What is an angle?
Ans: An angle is the union of two non-collinear rays with a common initial point.

Q.3. What is a reflex angle?
Ans: An angle whose measure is more than \({180^ \circ }\) is called a reflex angle. The measure of a reflex angle is more than \({180^ \circ }\) and less than \({360^ \circ }.\)

Q.4. What are all the angles measurements?
Ans: The different types of angle measurements depends on their measurements are:
1. Zero angle
2. Acute angle
3. Obtuse angle
4. Right angle
5. Straight angle
6. Complete angle

Q.5. What are the different types of units used in angle measurement?
Ans: The different types of angle measurement are:
1. Degrees
2. Revolution
3. Radian

Now you are provided with all the necessary information on the measurement of angles and we hope this detailed article is helpful to you. If you have any queries regarding this article, please ping us through the comment section below and we will get back to you as soon as possible.