## What is Angle?

An angle is formed when two straight lines or rays meet at a common endpoint. The common point of contact is called the vertex of an angle. The word angle comes from a Latin word named ‘angulus,’ meaning “corner.”

**Angles Around Us:**

There are many daily life examples of an angle, such as cloth-hangers, arrowheads, scissors, partly opened doors, pyramids, edge of a table, edge of a ruler, etc.

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**Symbol of Angle**

The symbol ∠ represents an angle. Angles are measured in degrees (°) using a protractor.

For example, 45 degrees is represented as 45°

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**Parts of Angles**

**Vertex:**A vertex is a corner of an angle, a point where two lines/sides meet. O is the vertex in the given figure.**Arms:**The two sides of the angle, joined at a common endpoint. OA and OB are arms of an angle.**Initial Side:**Also known as the reference line, a straight line from where an angle is drawn. OB is the reference line.**Terminal Side:**The side up to which the angle measurement is done. In the given diagram below, OA is the terminal side.

**Types of Angles**

Based on their measurements, here are the different types of angles:

- An
**acute angle**measures less than 90° at the vertex. - An
**obtuse angle**is between 90° and 180°. - A
**right angle**precisely measures 90° at the vertex. - An angle measuring exactly 180° is a
**straight angle**. - A
**reflex angle**measures between 180°- 360°. - A
**complete**angle measures 360°.

**Interior and Exterior Angles:**

Interior angles: Interior Angles are the angles formed within or inside a shape.

Here, ∠ABC, ∠BCA and ∠CAB are interior angles.

Exterior angles: Exterior angles are the angles formed outside a shape, between any side of a shape and an extended adjacent side. Here, ∠ACD is an exterior angle.

**Complementary and Supplementary Angles:**

Complementary angles: Angles that add up to 90° (a right angle) are called complementary angles.

Here, ∠BXC and ∠CXD are complementary angles.

Supplementary angles: Angles that add up to 180° (a straight angle) are called supplementary angles.

Here, ∠AXD and ∠CXD are supplementary angles.

**Real-life Application of Angles**

- Engineers construct buildings, bridges, houses, monuments, etc., using angle measurement.
- Athletes use its concept in sports to enhance their performance.
- Carpenters use it to make equipment like doors, chairs, sofas, tables, etc.
- Artists use their measurement knowledge to sketch or create art pieces.
- Wall clocks use the concept of angles to show time with hour and minute hands.

**How to Construct an Angle (using protractor)**

- Draw a ray OA of any length.
- Now, place the protractor at that point, and its midpoint should touch the marked point O.
- Now mark the point as B on the top circular part of a protractor, according to the preferred angle for example 40°.
- Draw a straight line joining those two points, O and B.
- Mark the degree of the angle made where two sides of the straight line intersect.

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## Solved Examples

### Example 1

Find missing angle x in the figure.

**Solution:**

We can see a ∠x + 35° = 90°

∠x = (90 – 35)° = 55°

### Example 2

Solve for x.

**Solution:**

5x – 70 = 105 (alternate angles)

5x = 175

Therefore, x = 35°

**Example 3**

In a triangle ABC, ∠A = 90 and ∠B = 30. Find ∠C?

**Solution:**

The sum of all 3 interior angles of a triangle is equal to 180°. Hence, 90° + 30° + x = 180°.

Solve for x

x = 180 – (90 + 30) = 60°.

## Practice Problems

## Angle1 ## The Sum of all angles around a point equals360° 180° 270° 90° CorrectIncorrect Correct answer is: 360° The Sum of all angles around a point equals 360° 2 ## Angles that sum up to 90° are known asvertical angles complementary angles reflective angles supplementary angles CorrectIncorrect Correct answer is: complementary angles Angles that sum up to 90° are known as complementary angles. 3 ## Angles that are opposite to each other are calledvertical angles complementary angles reflective angles supplementary angles CorrectIncorrect Correct answer is: vertical angles Angles that are opposite to each other are called vertical angles. |

**Frequently Asked Questions**

**Mention the types of angles based on the direction of a cycle?**

The angles are of two types based on the direction of the cycle:

- Positive Angles: Positive angles are measured in the anticlockwise or counterclockwise direction from the baseline. Positive angles are written with or without the plus sign before angles. It is drawn from the (+x, +y) plane.
- Negative Angles: Negative angles are measured clockwise from the baseline. It is drawn from the origin in the (+x, -y) plane.

**Do angles on a straight line always add up to 180°?**

- Angles sharing a vertex and one side of a line add up to 180°.
- Angles with a common vertex occupying the space around a point add up to 360°.

**What are the types of angles?**

Types of angles based on the measurement are:

- Right angle
- Straight angle
- Reflex angle
- Obtuse angle
- Acute angle

**What is the application of angles in math?**

Engineers and architects use angles for building, measurement, designing, etc. Architects and engineers use it to design roads, buildings, and sporting facilities.