Angles: Definition, Diagram, Types, Properties, Examples

Angles: Chairs, windows, mobile phones, books – these are some of the things that we see in our daily life. The inclination of the lines where they intersect is called an angle, whereas corners or vertices are formed when two lines or line segments intersect at a point. Angle is originated from the Latin word “Angulus”. There are many types of angles such as; Acute, Right, Obtuse, Straight, and Reflex angle. Angle is measured in degrees and radians. Students must go through this article to learn about the types of angles and their applications.

What is an Angle?

An angle is formed when two rays originate from the same point. The rays making an angle are called the arms of an angle and the originating point is called the vertex of an angle.

Angle is represented by the symbol \(\angle \). Here in the diagram, the angle formed is represented as \(\angle PQR\).

Uses of Angles

Angles are very important in our daily lives. Carpenters make use of angles in order to get proper measurements and structure of the furniture they make. One of the basic needs of our life is shelter; architects and engineers create plans and designs of buildings and houses using angles. They also use angles to design roads.

Coming to sports, athletes use angles to enhance their performance, such as steering the car, throwing a baseball, shooting the ball into the basket, kicking the soccer ball, etc.

Types of Angles

The different types of angles are mentioned below:

1. Acute angle
2. Obtuse angle
3. Right angle
4. Straight angle
5. Reflex angle
6. Complete/Full angle

Acute Angle:

An angle which is in between \(0^\circ \) to \(90^\circ \) is known as an acute angle. If all the angles of a triangle are less than \(90^\circ \) then they are known as acute triangles. In case of equilateral triangle, the measurement of each angle is \(60^\circ \). Therefore, equilateral triangles are acute triangles.

Obtuse Angle:

An angle which is more than or \(90^\circ \) but less than \(180^\circ \) is known as an

obtuse angle. In an obtuse triangle, only one angle will be obtuse and the remaining two angles will be acute angles. Hence, the sum of the angles in the triangle should be \(180^\circ \).

Right Angle:

An angle which is exactly of \(90^\circ \) is known as right angle. In a right triangle one angle is of \(90^\circ \). The longer side of the right triangle which is opposite to the right angle is known as hypotenuse.

Straight Angle:

When the arms of the angles lie in the opposite direction, they form a straight angle. That is, an angle of \(180^\circ \) is known as straight angle or flat angle.

One straight angle is a combination of two right angles.

Reflex Angle:

An angle which is greater than \(180^\circ \) but less than \(360^\circ \) is known as reflex angle. Reflex angle corresponding to interior angle of a triangle with a measure of \(115^\circ \) is \(360^\circ – 115^\circ \) or \(245^\circ \).

In the below diagram, the shown angle is represented as \(reflex\,\angle PQR\).

Complete/full Angle:

An angle which is of \(360^\circ \) is known as complete or full angle. It is equivalent to two straight angles or four right angles.

Complementary & Supplementary Angles

Complementary Angles:

Two angles whose sum is of \(90^\circ \) are known as complementary angles. Whenever two angles are said to be complementary, each of the angle is said to be the complement of the other. As shown in the diagram below, \(30^\circ \)angle is the complement of \(60^\circ \) angle or vice versa because their sum is \(90^\circ \).

Supplementary Angles:

Two angles whose sum is of \(180^\circ \) are known as supplementary angles. When two angles are supplementary, each angle is said to be the supplement of the other. As shown in the diagram below, \(60^\circ \)angle is the supplement of \(120^\circ \)angle or vice versa because their sum is \(180^\circ \).

Angles Formed by a Transversal

When two parallel lines are cut by a transversal, eight angles are formed by their intersection as shown in the diagram below.

Now by looking at the above diagram, we will write the angles on the basis of a relative position they have taken. The different angles formed are as below:

Corresponding Angles:

Corresponding angles are the angles which are formed by matching corners or corresponding corners with transversal when two parallel lines are intersected by any other line and they are always equal in measure. The pair of corresponding angles given in the below diagram are \(\angle 1\) and \(\angle 5\), \(\angle 2\) and \(\angle 6\), \(\angle 3\) and \(\angle 7\), \(\angle 4\) and \(\angle 8\).

Alternate Interior Angles:

The alternate interior angles are positioned at the inner corners of the intersections and they lie on the opposite sides of the transversal and they are always equal in measure. The pair of angles of alternate interior angles are \(\angle 3\) and \(\angle 6\), and the second one is \(\angle 4\) and \(\angle 5\).

Alternate Exterior Angles:

Alternate exterior angles are the pair of angles that lie on the outside of the two parallel lines but on the either side of the transversal line and they are always equal in measure. The pair of angles of alternate exterior angles are \(\angle 1\) and \(\angle 8\), and the other one is \(\angle 2\) and \(\angle 7\).

Co-Interior Angles:

The angles which lie on the same side of the transversal line and are positioned at the inner corners like the angles \(\angle 3\) and \(\angle 5\) and the angles \(\angle 4\) and \(\angle 6\) are co-interior angles and their sum is \(180^\circ \).

Co-Exterior Angles:

The angles which lie on the same side of the transversal line and are positioned at the exterior of the parallel lines like the angles \(\angle 1\) and \(\angle 7\) and the angles \(\angle 2\) and \(\angle 8\) are co-exterior angles and their sum is \(180^\circ \).

Vertically Opposite Angles:

When the angles are opposite to each other they are known as vertically opposite angles, like the angles \(\angle 1\) and \(\angle 4\), \(\angle 2\) and \(\angle 3\), \(\angle 5\) and \(\angle 8\) and the angles \(\angle 6\) and \(\angle 7\), as shown in the diagram below.

How to Measure an Angle?

The standard unit to measure an angle is degree. The easiest way to measure an angle is by using a protractor. While using the protractor to measure the angle, start by lining up one ray along the (0^\circ ) line on the protractor, then line up the vertex to the midpoint of the protractor. Later, follow the second ray to determine the angle’s measurement to the nearest degree.

Angles of real objects are measured using different tools like an inclinometer or clinometer. Another unit for angle measurement is the radian.

What are Congruent Angles?

Angles which are having exactly the same measures are known as congruent angles. Now, in the given diagram below \(\angle P\) is congruent to \(\angle Q\) as the degree measure for both is \(110^\circ \).

Two or more triangles are said to be congruent when their corresponding angles or sides are equal. Congruent triangles have the same shape and same dimensions and the symbol to represent congruency of triangles is \(\cong \).

Solved Examples of Angles

Q.1. Find the complement of the angle: \(60^\circ \)
Ans: The given angle is \(60^\circ \).
Let the measure of its complement be \(x^\circ \). Then,
\(x + 60^\circ = 90^\circ \Rightarrow x = \left({90^\circ – 60^\circ } \right) = 30^\circ \)
Hence, the complement of the given angle measures \(30^\circ \).

Q.2. Find the supplement of the angle: \(125^\circ \)
Ans: The given angle measures \(125^\circ \)
Let its supplement be \(x^\circ \). Then,
\(x + 125^\circ = 180^\circ \Rightarrow x = \left({180^\circ – 125^\circ } \right) = 55^\circ \)
Hence, the supplement of the given angle measures \(55^\circ \).

Q.3. In the adjoining figure what value of \(x\) will make \(AOB\) a straight line?

Ans: \(AOB\) will be straight line, if \(\angle AOC + \angle BOC = 180^\circ \).
Therefore, \((3x + 5^\circ ) + (2x – 25)^\circ = 180^\circ \Rightarrow (5x – 20)^\circ = 180^\circ \)
\( \Rightarrow 5x = (180 + 20)\)
\( \Rightarrow 5x = 200 \Rightarrow x = 40^\circ\)
Hence, \(x = 40^\circ\) will make \(AOB\) a straight line.

Q.4. Find the angle which is its own complement.
Ans: Let the measure of the required angle be \(x\). Then,
\(x + x = 90^\circ \Rightarrow 2x = 90^\circ \Rightarrow x = 45^\circ \)
Hence, the required angle measures \(45^\circ \).

Q.5. How many right angles make the angle \(360^\circ \)?
Ans: We know, \(90^\circ \) is equivalent to \(1\) right angle.
Hence, \(360^\circ \) is equivalent to \(\frac{{360^\circ }}{{90^\circ }} = 4\) right angles.

Summary

Through this article, we have learned that an angle is a geometric figure formed using two rays. The types of angles we have studied are zero angles, acute angle, obtuse angle, right angle, reflex angle, straight angle, complete angle, complementary angles, supplementary angles, transversal angles, etc. The tools and ways to measure the angles are also covered, besides the unit of angles.

Angles that are formed by transversals are corresponding angles, alternate interior angles, alternate exterior angles, co-interior angles, co-exterior angles, and vertically opposite angles. We also discussed that alternate angles are equal and exterior angles on the same side of transversals are supplementary.

FAQs on Angles

Q.1. Do congruent angles add up to \(180^\circ \)?
Ans: If the given angles are congruent then their measures of the angles will be the same. Hence, only when both the angles are \(90^\circ \), then the sum would be \(180^\circ \).

Q.2. What are angles?
Ans: An angle is formed when two rays originate from the same originating point. The rays making an angle are called the arms of an angle and the originating point is called the vertex of an angle.

Q.3. What are 5 types of angles?
Ans: There are many types of angles and such types of angles are acute angle, obtuse angle, right angle, straight angle, and reflex angle.

Q.4. What are the angles based on rotation?
Ans: There are two types of angles based on rotation, they are:
Positive angles and Negative angles

Q.5. What is a zero angle?
Ans: An angle that is equal to \({0^\circ }\) is known as a zero angle.

Q.6. What is the tool to measure the angle?
Ans: The easiest and the common way is to use the protractor to measure the angles.

Q.7. What is the sum of angles in the parallelogram?
Ans: The sum of all the angles in a parallelogram is equal to \(360^\circ\).

Q.8. What are adjacent angles?
Ans: The two angles which have a common vertex and a common side but they don’t overlap are known as adjacent angles.

Q.9. What are vertical angles?
Ans: When two lines intersect, then the angles which are formed opposite to each other are known as vertical angles. Vertically opposite angles are always equal.

Q.10. Are the interior angles on the same side of transversal congruent?
Ans: Two interior angles on the same side of a transversal are congruent in the special case when the transversal intersects the pair of parallel lines perpendicularly. But, in all other cases, where the transversal does not intersect the pair of parallel lines perpendicularly, they are not congruent.

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