An angle is a convergence of two rays or lines with a common endpoint. The common endpoint is the vertex of an angle, and the rays are the sides or arms of an angle. If O is the vertex of an angle while A and B are points on the two sides, the angle may get a reference as ∠AOB or ∠BOA, or only ∠O. Note that the vertex letter is always placed in the centre if it follows a three-letter notation.

In elementary geometry, this definition of angle works. Although to understand the vastness of the topic, it is better to study about different types of angles. It is interesting to note that, just as line segments, angles can be compared, added, and even subtracted. So, this definition works to a certain extent; however, one has to look at the other means to define angles in detail.

Another definition states an angle as - The amount of rotation about the point of intersection of two planes or lines, which is required to bring one in correspondence with the other is called an angle.

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Let us elaborate further. Ref fig.1 above.

Vertex- The vertex is the single point at which the two lines or rays join together. Point O is the vertex of the ∠AOB.

Arms, Sides or Legs- The legs of an angle are the two lines that make it up. In the above figure, line segments AO and OB are the legs, arms, or sides of the ∠AOB.

Interior – The interior of an angle is the space between the two sides that extend out to infinity.

Exterior – The entire space on the plane that is not in the interior at an angle.

There are different types of angles. It is easy to define and compare angle by measurements of their degrees. The standard terms that are in use to measure angles are degree (°), radians, or gradians. Commonly, the term degrees is in use to determine and classify the angles. Ref Fig.2 below

1 - Acute angle is 0 degree to 90 degrees, excluding both.

2 – Obtuse angle is 90 degree to 180 degrees, excluding both.

3 – Right angle that is exactly 90 degrees

4 – Straight angle that is exactly 180 degrees.

5 – Reflex angle that is 180 degree to 360 degrees, excluding both.

6 – Full rotation angle that is exactly 360 degrees.

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In trigonometry, when talking about different types of angles, it is essential to remember that angles have some additional properties. They can have a measure greater than 360°, can be positive (anti-clockwise direction) or negative (clockwise direction), and are also positioned on a coordinated grid or graph, with x and y axes. Angles are usually measured in radians instead of degrees in trigonometry.

As you know, the most common angles are 90 degrees, 180 degrees, and 360 degrees. Here, we will study about complementary angles. As mentioned earlier, angles can be added as well as subtracted. When it comes to a right angle or 90° angle, two angles can add up to make a 90-degree angle. For example, a 40° and 50° angle are complementary angles as they add up to 90°.

Complementary comes from Latin completum. In aright triangle, two smaller angles are always complementary. Ref. Fig. 2a and Fig.2b below

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When talking about complementary angles, remember that they are always in pairs. Referring to one of the angles in a complementary pair, we can say that one angle is the complement of the other.

You already know that an angle gets a measurement in degrees. When we say that an angle ABC, we mean the actual angle. When we want to talk about the size or measure of the angle in degrees, we report the measurement of the angle ABC. The measure of an angle is always written as m ∠ ABC. Many times, we write ∠ ABC=45°. It is not a correct method- it should be written as m∠ ABC= 45°.

FAQ (Frequently Asked Questions)

Q1. Can Three Angles be Complementary?

First of all, it is essential to know that complementary angles are a pair of angles that add up to 90 degrees. Three angles cannot be complementary. Three or more angles are not called complementary angles, even if their measures add up to 90 degrees. Complementary angles are always in a pair. Two angles measuring 45 degrees are complementary, and there can be more such pairs that add up to 90 degrees. A pair of angles can be in any measurement of degrees, such as 30 and 60, 35 and 55, but to be complementary they have to be in a pair.

Q2. Are Complementary Angles Congruent?

Complementary angles are those pairs of angles whose sum is 90 degrees. In other words, it is not necessary that a complementary angle should always be congruent. These angles can be placed, so as to form perpendicular lines, or they may be two separate angles. So, we can say that the angles that complement similar angles or congruent angles are congruent. For example, two complementary angles to be congruent that have to have the same measure of 45 degrees each. As you know that congruent angles are those angles that are equal or have the same degrees of measurement.