Angles at a Point: Definition, Diagram, Types, and Examples

An angle is formed when the two rays are combined at a common point. The common point here is the vertex, and the two rays are known as the arms of the angle. The symbol ′\(\angle \)′ represents the angle. In this article, we shall discuss the angles and their definition, various types of angles, angles around a point, angles made by a straight line, etc.

What is an Angle?

Angles are formed when lines or line segments meet. A corner or a vertex is formed when two lines or line segments intersect at a point. In our daily life, you get to see various angles formed between the edges of the plane surfaces. To make a similar kind of model using the plane surface, you need to have a thorough knowledge of angles.

Definition: An angle is formed when the two rays originate from the same point. The rays making an angle are known as the arms of an angle, and the originating point is known as the vertex of an angle.

The symbol represents the angle \(\angle \).  Here, in the diagram, the angle formed is represented \(\angle PQR\).

Parts of an Angle

In the geometry, the angle is formed when the two rays are joined at their endpoints, and there are various parts of the angles that are given below:

1.Sides of the angles: The two rays are the sides of the angles.
2. Vertex: The angle that has a common endpoint shared by the two rays is the vertex.
3. And the angle is always measured in the degree. One complete rotation is equal to \({360^ \circ }\)
4.Size of the angle: An easy way to measure an angle is to use the protractor, and the standard protractor’s size is \({180^ \circ }\). There are two sets of numbers on the protractor, and they are:

a. One is the clockwise direction, and the other one is
b. Anti-clock direction

Types of Angles

We have different types of angles which are written below:

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

Zero Angles: \(PQ\) and \(PR\)  are two rays. The given rays’ starting points are on the left side, and both of them overlap. In such a situation, the angle created between the rays \(PQ\) and \(PR\) represents the zero angle. The same is given in the below diagram.

Acute Angle: An angle 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 such a triangle is known as an acute triangle. In the case of an equilateral triangle, each angle is \({60^ \circ }\) so equilateral triangles are acute triangles.

Obtuse Angle: An angle that 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 then the remaining two angles will be acute angles such that the sum of the angles in the triangle should be \({180^ \circ }\).

Right Angle: An angle that is exactly \({90^ \circ }\) is called the right angle. In a right triangle, one angle is \({90^ \circ }\) The longer side of the right triangle that is opposite to the right angle is known as hypotenuse.

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

One straight angle is the combination of the two right angles.

Reflex Angle: An angle greater than \({180^ \circ }\) but less than \({360^ \circ }\) is called the reflex angle. Reflex angle corresponding to the interior angle of a triangle with a measure of \({115^ \circ }\) is \({115^ \circ }\) is \({360^ \circ } – {115^ \circ }\) or \({245^ \circ }\).

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

Complete/full angles: An angle of \({360^ \circ }\) is called the complete angle. It is the equivalent of two straight angles or four right angles.

Complementary Angles: The two angles whose sum is \({90^ \circ }\) are called complementary angles. Whenever the two angles are known to be complementary, each angle is said to complement 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: The two angles whose sum is \({180^ \circ }\) are called supplementary angles. When the two angles are supplementary, each angle is known 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 Around a Point

Angles formed by several rays having a common initial point are called angles at a point.

In the given diagram, the rays \(OA,\,OB,\,OC,\,OD\,\) having a common initial point \(O,\,\) form \(\angle 1,\,\angle 2,\,\angle 3,\angle 4\) at the point \(O,\,\)

If you find the measures of these angles, you will find that

\(\angle 1\, + \angle 2 + \,\angle 3 + \angle 4 = {360^ \circ }\)

Thus, the sum of all the angles at a point is \(4\) right angles or \({360^ \circ }\).

Sum of all Angles Around a Point

An angle is measured concerning the circle with its centre at the common endpoint of rays. Therefore, the sum of the angles at the point is always \(360\)  degrees.

Angles at a Point add up to 360 degrees

Angles around a point add up to \({360^ \circ }\) his can be used to calculate the missing angles.

Example: Calculate the angle \(a\)

\(a = \,\,{360^ \circ } – {60^ \circ }\, – \,{95^ \circ }\, – \,{105^ \circ } = {100^ \circ }\)

Angles on the straight line

They always add up to \({180^ \circ }\).  This fact can also be utilised to calculate the angles.

Example: Calculate the angles \(b\) and \(c\).

\(b = {180^ \circ } – {30^ \circ } = {150^ \circ }\)

\(c = {180^ \circ } – {150^ \circ } = {30^ \circ }\)

\(c\) and \({30^ \circ }\)  are known as the vertically opposite angles.

Hence, vertically opposite angles are equal.

Solved Examples – Angles of a Point

Q.1. Let’s measure the complement of each of the given angles: \({60^ \circ }\)
Ans: The given angle is \({60^ \circ }\)
Let the measure of its complement be \({x^ \circ }\), Then,
\(x + 60\, = 90\)
\( \Rightarrow x(90 – 60)\, = 30\)
Therefore, the complement of the given angle measures is \({30^ \circ }\).

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

Q 3. In the adjoining figure, what value of \(x\) will make \(AOB\) a straight line?
Ans: \(AOB\) will be a straight line if \(\angle AOC + \angle BOC = 180^\circ \)
Therefore, \((3x + 5) + (2x – 25) = 180 \Rightarrow 5x – 20 = 180\)
\( \Rightarrow 5x = (180 + 20)\)
\( \Rightarrow 5x = 200\) \( \Rightarrow x = 40\)
 Hence, \(x = 40\) will make \(AOB\) a straight line.

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

Q 5. Identify the least number, when multiplied by \({40^ \circ }\) angles, make it the reflex angle?
a) \(2\)
b) \(5\)
c) \(6\)
d) \(7\)
Ans: We know that,
a) \(2 \times {80^ \circ } = {80^ \circ },\) that is between \({0^ \circ }\) and \({90^ \circ }\) so acute angle.
b) \(5 \times {205^ \circ } = {80^ \circ },\)  that is between \({180^ \circ }\) and \({360^ \circ }\) so reflex angle.
c) \(6 \times {240^ \circ } = {80^ \circ },\) that is between \({180^ \circ }\) and \({360^ \circ }\)  so reflex angle.
d) \(7 \times {40^ \circ } = {285^ \circ },\) that is also reflex,
We observe that each of the numbers \(5,\,6\) and \(7\) when multiplied by \({40^ \circ }\)  makes it a reflex angle. But, \(5\) is the least among them.
Hence,\(5\) is the required answer.

Summary

In the given article, you can see we have discussed the angle of a point followed by the definition of angles. Then we talked about parts of the angles, later discussed types of the angles that included acute angle, obtuse angle, straight angle, zero angles, complementary angle, supplementary angle, straight, and complete angle. We glanced at angles around a point, then the sum of all the angles around a point is known as, later talked about angles at a point add up to \({180^ \circ }\) Then we have provided solved examples along with a few FAQs.

Frequently Asked Questions (FAQs) – Angles of a Point

Q 1. How do you find the angle at a point?
Ans: Angles formed by a number of rays having a common initial point are called at a point. In the given diagram, the rays \(OA,\,OB,\,OC,\,OD\) having a common initial point \(O\), form \(\angle 1,\angle 2,\angle 3,\angle 4\) at the point \(O\).
If you find the measures of these angles, you will find that
\(\angle 1 + \angle 2 + \angle 3 + \angle 4 = 360^\circ \)
Thus, the sum of the measures of all the angles at a point is \(4\) right angles or \({360^\circ }\).

Q 2. What do angles at a point mean in math?
Ans: The angles at a point in the figure formed by the two rays are known as the sides of the angle, which share the common endpoint, known as the vertex of the angle.

Q 3. What are the types of angles?
Ans: There are various types of angles, and those are zero angle, acute angle, obtuse angle, right angle, straight angle, reflex angle and complete angle.

Q 4. Do all angles of a triangle equal \({180^\circ }\)?
Ans: Yes, the angles of the triangle add up to \({180^\circ }\)

Q 5. What is the sum of all the angles around a point?
Ans: An angle is measured with reference to the circle with its centre at the common endpoint of rays. Therefore, the sum of the angles at the point is always \(360\) degrees.

Learn about Triangles here

We hope you find this article on ‘Angles at a Point‘ helpful. In case of any queries, you can reach back to us in the comments section, and we will try to solve them.