Angle Meaning
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.
Parts of 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.
Types of Angles
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.
Complementary Angles
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.
Measure of an Angle
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°.
FAQs on Angles and It's Types
1. What are the main types of angles based on their measurement?
Angles are classified into several main types based on their degree measure. The most common types are:
- Acute Angle: An angle that measures less than 90°.
- Right Angle: An angle that measures exactly 90°.
- Obtuse Angle: An angle that measures more than 90° but less than 180°.
- Straight Angle: An angle that measures exactly 180°, forming a straight line.
- Reflex Angle: An angle that measures more than 180° but less than 360°.
- Complete Angle: An angle that measures exactly 360°.
2. What is the difference between complementary and supplementary angles?
The main difference lies in their total sum. Complementary angles are a pair of angles that add up to 90 degrees. For example, 30° and 60° are complementary. Supplementary angles are a pair of angles that add up to 180 degrees. For example, 110° and 70° are supplementary.
3. What defines an adjacent angle and a linear pair?
Adjacent angles are two angles that share a common vertex and a common side, but do not overlap. A linear pair is a special type of adjacent angle pair where their non-common sides form a straight line. The sum of angles in a linear pair is always 180°, making them supplementary.
4. What is the relationship between vertically opposite angles?
When two lines intersect, they form two pairs of vertically opposite angles. The key property of these angles is that they are always equal to each other. For instance, if one angle is 120°, the angle vertically opposite to it will also be 120°.
5. How can corresponding and alternate interior angles prove that two lines are parallel?
When a third line, called a transversal, intersects two other lines, specific angle pairs are formed. You can prove the two lines are parallel if:
- A pair of corresponding angles is equal.
- A pair of alternate interior angles is equal.
If either of these conditions is met, it confirms that the two lines are parallel to each other.
6. Is it necessary for complementary or supplementary angles to be adjacent?
No, it is not necessary. While adjacent angles can be complementary or supplementary (like in a linear pair), the terms simply refer to the sum of the angles. For example, a 30° angle in one corner of a room and a 60° angle in another are still complementary. Similarly, a 100° angle and an 80° angle are supplementary regardless of their position.
7. Can two obtuse angles ever be supplementary? Why or why not?
No, two obtuse angles can never be supplementary. An obtuse angle is defined as being greater than 90°. The sum of two angles both greater than 90° will always be more than 180°. Since supplementary angles must add up to exactly 180°, it is impossible to form a supplementary pair with two obtuse angles.
8. What are some real-world examples where different types of angles are important?
Angles are fundamental in many real-world applications:
- Architecture and Construction: Right angles (90°) are essential for creating stable corners in buildings. Roof pitches are designed using acute and obtuse angles.
- Navigation: Pilots and sailors use angles to plot courses and determine headings.
- Art and Design: Artists use angles to create perspective and composition.
- Sports: Athletes, like a pool player or a basketball shooter, intuitively use angles to aim and score.
