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The ancient Egyptian and Mesopotamian civilizations introduced the concept of angles as an early system of measurement and geometry. They used angles to analyze the land, build up buildings, design roads, etc.

Then Greek scientists or mathematicians such as Thales, Pythagoras, and Euclid made a very important addition to the study of angles. Pythagoras’ theorem is known as the backbone of geometry and everyone is familiar with Euclidean geometry. These are the great contributions of Greek mathematicians.

An angle is a shape / a figure formed by two rays when these rays contribute at a general endpoint known as the **vertex of an angle**. An angle is a measurement between different rays in the form of a **radian **or **degree**.

- Ray
**A**is an upper-arm - Ray
**C**is a lower-arm - The endpoint “
**B**” where both rays meet is known as the vertex. - The angle is a gap or a space between two lines that meet at a common point known as a vertex.

Different types of angles are as follows:

A type of angle in which the degree is always less than** 90**^{o }is known as an acute angle. An acute angle is always sharp and narrow because the “**space**” which is known as the angle is very less between two rays.

A type of angle in which the degree is always equal to **90 ^{o}** is known as the

A type of angle having such a degree that lies between **90 ^{o }**and

A type of angle in which the degree is** 180 ^{o}** is known as the

A type of angle which have a degree greater than **180**^{o }and less than **360 ^{o}** is known as the

A type of angle in which the degree is **360 ^{o}** is known as the

A pair of angles that share a common vertex or have the same endpoint is known as the **adjacent angle**. The common side made a line segment that connects the vertices of the adjacent angle.

A pair of angles whose sum is **90 ^{o}** is known as the

A pair of angles whose sum is **180 ^{o}** is known as the

Some applications of angle are as follows:

- Robotics and automation
- Computer graphics and animations
- Physics and mechanics
- Navigation and survey
- Engineering and Architecture
- Geometry and trigonometry

These are some applications of angles where angle plays a constructive role. Angle plays a very important role in all the above-mentioned fields.

**Example 1: **

In triangle âˆ†ABC, ∠A = 80^{o}, and **∠****B = 60 ^{o} find **

**Solution:**

**Step 1:** Add the ∠A, ∠B, and **∠****C** which is equal to 180^{o}.

∠A + ∠B + **∠****C** = 180^{o}

**Step 2: **Put the values of ∠A and ∠B.

80^{o} + **60 ^{o }**+

**Step 3: **Simplification.

140^{o }+ **∠****C = 180 ^{o}**

**∠****C = 180 ^{o} - **140

**∠****C = 40 ^{o}**

**Example 2:**

Determine the value of two angles A and B that are complementary, given that ∠A = (4x – 30)^{ o} & ∠B = (3x – 20)^{ o}.

**Solution:**

**Step 1:** By adding **∠A **and **∠B** it is equal to **90**^{o }as it is a complementary angle.

∠A + ∠B = 90^{o}

**Step 2:** Substituting the values of **∠A **and **∠B**.

(4x – 30)^{ o} + (3x – 20)^{ o} = 90^{o}

**Step 3: **Simplification.

(7x – 50)^{ o }= 90^{o}

7x = 90^{o }+ 50^{o}

7x = 140^{o}

x = 140^{o} / 7

x = 20^{o}

**Step 4:** Put the value of **“x”** in **∠A**.

∠A = (4x – 30)^{ o}

∠A = (4(20^{o}) – 30)^{ o}

∠A = 80^{o} – 30^{o}

∠A = 50^{o}

**Step 5:** Put the value of “**x**” in **∠B**.

∠B = (3x – 20)^{ o}

∠B = (3(20^{o}) – 20)^{ o}

∠B = 60^{o} – 20^{o}

∠B = 40^{o}

**Example 3: **

Determine the value of two angles C and D that are supplementary, given that **∠C = (3y – 40) ^{ o} & ∠D = (2y – 30)^{ o}**.

**Solution:**

**Step 1:** By adding **∠C** and** ∠D **it is equal to **180**^{o }as it is a supplementary angle.

∠C + ∠D = 180^{o}

**Step 2:** By putting the values of **∠C** and **∠D.**

(3y – 40)^{ o} + (2y – 30)^{ o} = 180^{o}

**Step 3: **Simplification.

(5y – 70)^{ o }= 180^{o}

5y = 180^{o }+ 70^{o}

5y = 250^{o}

y = 250^{o} / 5

y = 50^{o}

**Step 4:** Put the value of “**y**” in **∠C**.

∠C = (3y – 40)^{ o}

∠C = (3(50^{o}) – 40)^{ o}

∠C = 150^{o} – 40^{o}

∠C = 110^{o}

**Step 5: **Put the value of “**y**” in **∠D**.

∠D = (2y – 30)^{ o}

∠D = (2(50^{o}) – 30)^{ o}

∠D = 100^{o} – 30^{o}

∠D = 70^{o}

In this article, we explain the angle in such a way that everyone can easily grab the basic concept of angle. An angle plays a very constructive role in the field of engineering, and in physics. Trigonometry is a major branch of mathematics and without these angles, it is considered nothing.

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