# Angles – Definition with Examples

## 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.

## Symbol of Angle

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

For example, 45 degrees is represented as 45°

## 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)

1. Draw a ray OA of any length.
2. Now, place the protractor at that point, and its midpoint should touch the marked point O.
3. Now mark the point as B on the top circular part of a protractor, according to the preferred angle for example 40°.
4. Draw a straight line joining those two points, O and B.
5. 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

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.
• 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°.

Types of angles based on the measurement are:

• Right angle
• Straight angle
• Reflex angle
• Obtuse angle
• Acute angle

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