## Angles

## What are Angles?

**Angles are a fundamental concept in geometry, representing the figure formed by two rays (called the sides of the angle) sharing a common endpoint, known as the vertex.** They are typically measured in degrees or radians and describe the amount of turn between each arm. Angles are used to describe the shape, size, and direction of corners found throughout daily life and various fields such as mathematics, engineering, and architecture.

## Parts of an Angle

An angle primarily consists of two essential parts:

**Arms**: These are the two rays that extend from a common point and form the angle. The arms can be of any length and direction but share the vertex as their starting point.

**Vertex**: This is the specific point at which the two arms meet. It acts as the pivotal point from which the angle is measured.

## Measure of an angle

An angle is measured in degree, with a full rotation around a point forming a complete angle of 360Ā°.

The most effective tool for measuring an angle is a protractor, which is typically a semi-circular, translucent instrument. It facilitates precise measurement of angles in degrees. A protractor is marked with degrees from 0Ā° to 180Ā° on both the outer and inner scales. The outer scale measures degrees clockwise, while the inner scale measures them counterclockwise. This dual scale system allows for easy and accurate reading of both acute and obtuse angles, making the protractor indispensable in geometry and various practical applications.

## Types of Angles and their Properties

Angles are categorized based on their measure, which defines their unique characteristics. Let’s explore each type of angle and its properties:

### Acute Angle

An **acute angle** measures greater than 0Ā° but less than 90Ā°. These angles are sharp and are often seen in various geometric figures.

### Right Angle

A **right angle** measures exactly 90Ā°, resembling the shape of the letter L. This type of angle is prevalent in everyday objects and represents a quarter turn.

### Obtuse Angle

An **obtuse angle** is one that measures more than 90Ā° but less than 180Ā°. These angles appear wider and are less common than acute angles.

### Straight Angle

A **straight angle** measures exactly 180Ā°, forming a straight line. This angle represents a half turn, consisting of two right angles back-to-back.

### Reflex Angle

A **reflex angle** measures more than 180Ā° but less than 360Ā°. These angles are larger than a straight angle and span more than half a turn.

### Complete Angle

A **complete angle** measures exactly 360Ā°, representing a full rotation around a point.

## Angle Measurement Based on Rotation

Angles can also be classified based on the direction of their rotation:

### Positive Angles

**Positive angles** are measured in a counterclockwise direction from the base. This is the standard direction for measuring angles in mathematics.

### Negative Angles

**Negative angles** are measured in a clockwise direction from the base. These angles represent a reverse rotation compared to the typical angle measurement.

## How to Measure an Angle?

We use protractors to accurately measure angles. Consider the angle ā AOB depicted in the figure below. At first glance, ā AOB appears to be an acute angle, indicating that its measurement is greater than 0Ā° and less than 90Ā°. Let’s explore how to measure this angle using a protractor effectively.

## How to Measure an Acute Angle?

Let us try to measure the given ā AOB.

**Step 1:** Position the protractor so that the ray OB aligns with the baseline of the protractor. Begin by observing the inner scale, starting from the 0Ā° mark located at the bottom right of the protractor. This alignment ensures accurate measurement of the angle.

**Step 2**: Identify the number on the protractor that aligns with the second ray to determine the angle’s measurement. Use the protractor’s inner scale for this measurement. Consequently, the angle ā AOB measures 37Ā°.

## How to Measure an Obtuse Angle?

Now, let us try to measure the given ā AOC.

**Step 1**: To measure the angle, use the outer scale of the protractor, starting from the 0Ā° mark located at the bottom left.

**Step 2**: The number on the outer scale of the protractor that aligns with ray OA indicates the measurement of ā AOC. Therefore, ā AOC measures 143Ā°.

## How to Construct Angles?

Constructing angles accurately is essential in various fields such as drafting, architecture, and even school projects. Hereās a step-by-step guide on how to draw a 50Ā° angle using a protractor:

Step 1: First, draw a straight line, labeling it as ray OB. Position the protractor so that its baseline aligns perfectly with ray OB, ensuring the center point of the protractor is at the endpoint of the ray, typically marked as O.

**Step 2:**Ā Using the inner scale of the protractor, mark a point A above the marking on the protractor that corresponds to 50Ā°.

**Step 3: **After removing the protractor, draw a ray starting at point O and extending through point A. This creates the desired angle, ā AOB, which measures 50Ā°.

## Important Notes on Angles

Here are some crucial points to understand about angles, which are essential for grasping their properties and applications in geometry:

**Acute Angle**: An angle that measures greater than 0Ā° and less than 90Ā°. These angles are smaller than a right angle and commonly found in various geometric shapes.**Right Angle**: Exactly measures 90Ā° and is often represented by the corner of a square or rectangle. This angle forms the shape of the letter L and is a fundamental aspect of perpendicular lines.**Obtuse Angle**: An angle that is greater than 90Ā° but less than 180Ā°. These angles are larger than a right angle and appear more ‘spread out.’**Straight Angle**: Exactly measures 180Ā°, appearing as a straight line. This angle signifies a half-turn and is equivalent to two right angles put together.**Reflex Angle**: An angle that measures greater than 180Ā° but less than 360Ā°. These angles extend more than a straight angle but less than a complete rotation.**Complete Angle**: Measures exactly 360Ā°, representing a full rotation around a point. This is also known as a perigon angle.**Complementary Angles**: Two angles whose sum equals 90Ā°. These are typically acute angles that pair up to form a right angle.**Supplementary Angles**: Two angles whose measures add up to 180Ā°. These can be two right angles or any combination of angles that reach a straight angle when added together.**Adjacent Angles**: Two angles that share a common vertex and side but do not overlap. These angles are next to each other and often appear in geometric constructions.**Vertical Angles**: Also known as opposite angles, these are equal angles formed by two intersecting lines. Vertical angles are always congruent.

## Practice Problems On Angles

### Problem 1: Identifying Angles

Draw the following angles and identify them as acute, obtuse, right, straight, reflex, or complete:

- An angle of 25Ā°
- An angle of 90Ā°
- An angle of 175Ā°
- An angle of 270Ā°
- An angle of 360Ā°

### Problem 2: Measuring and Drawing Angles

Using a protractor, measure and draw the following angles:

- Draw an acute angle of 45Ā°.
- Draw a right angle.
- Draw an obtuse angle of 120Ā°.
- Draw a straight angle.
- Draw a reflex angle of 300Ā°.

### Problem 3: Angle Calculation

Calculate the missing angle in each scenario:

- Two angles in a triangle measure 65Ā° and 35Ā°. What is the measure of the third angle?
- If two angles on a straight line add up to 180Ā° and one angle is 110Ā°, what is the measure of the other angle?
- An angle is three times the size of its complement. What are the measures of the two angles?

### Problem 4: Real-Life Application

Identify the type of angle formed in the following real-life situations:

- The angle between the hands of a clock at 3:00.
- The angle formed by the open lid of a laptop partially closed at about halfway.
- The angle formed by a slice of pizza cut with a straight line from the tip to the crust edge.

### Problem 5: Creating Complex Figures

Create a diagram that includes the following:

- Two acute angles that are complementary.
- One obtuse angle adjacent to a right angle.
- A reflex angle paired with an acute angle in a scenario of your choice.

## Solved Examples On Angles

### Example 1: Identifying Angle Types

**Question**: Identify the type of each angle:

- An angle of 30Ā°
- An angle of 95Ā°
- An angle of 180Ā°
- An angle of 270Ā°

**Solution**:

**30Ā°**is an**acute angle**because it is less than 90Ā°.**95Ā°**is an**obtuse angle**because it is greater than 90Ā° but less than 180Ā°.**180Ā°**is a**straight angle**because it is exactly 180Ā°.**270Ā°**is a**reflex angle**because it is greater than 180Ā° but less than 360Ā°.

### Example 2: Calculating Missing Angle in a Triangle

**Question**: A triangle has two angles measuring 45Ā° and 70Ā°. What is the measure of the third angle?

**Solution**: The sum of the angles in a triangle is always 180Ā°. Therefore: Third Angle=180Ā°ā(45Ā°+70Ā°)=180Ā°ā115Ā°=65Ā°Third Angle=180Ā°ā(45Ā°+70Ā°)=180Ā°ā115Ā°=65Ā° The third angle measures **65Ā°**.

### Example 3: Complementary and Supplementary Angles

**Question**: If one angle measures 65Ā°, find its complementary and supplementary angles.

**Solution**:

**Complementary angle**: 90Ā°ā65Ā°=25Ā°90Ā°ā65Ā°=25Ā° The complementary angle is**25Ā°**.**Supplementary angle**: 180Ā°ā65Ā°=115Ā°180Ā°ā65Ā°=115Ā° The supplementary angle is**115Ā°**.

### Example 4: Using Vertical Angles

**Question**: Two intersecting lines form four angles. If one of the angles is 110Ā°, what are the measures of the other three angles?

**Solution**:

- The angle opposite the 110Ā° angle is also
**110Ā°**(vertical angles are equal). - The angles adjacent to the 110Ā° angles must add up to 180Ā° since they are supplementary. 180Ā°ā110Ā°=70Ā°180Ā°ā110Ā°=70Ā°
- Therefore, the other two angles are each
**70Ā°**.

## FAQs

## What is the definition of angle?

An angle is a geometric figure formed by two rays sharing a common endpoint called the vertex. It measures the rotation required to align one ray with the other.

## What is called the angle?

The angle refers to the space between two intersecting lines or surfaces at or close to the point where they meet, usually measured in degrees.

## What are 7 angles?

The seven types of angles are acute, right, obtuse, straight, reflex, complete, and perigon angles, each defined by its unique range of measurements.

## What is an angle slang?

In slang, “angle” refers to a calculated approach or perspective, often used to imply someone has a specific motive or is looking for a particular advantage in a situation.

## What is the difference between angel and angle definition?

“Angel” refers to a spiritual being believed to act as a messenger of God, typically depicted with wings. “Angle” is a mathematical term describing the space between two intersecting lines or surfaces.

## What is an angle definition for kids?

For kids, an angle can be described as a shape made when two lines meet at a point. You can think of it as the corner of a piece of paper or a book.

## How do you explain angles to a child?

Explain angles to a child by demonstrating how two sticks or pencils can form a space or opening when they meet at one end. The size of the opening helps us understand how big or small the angle is.

## What is every angle called?

Every angle is identified based on its measurement: acute (less than 90Ā°), right (exactly 90Ā°), obtuse (between 90Ā° and 180Ā°), straight (180Ā°), reflex (between 180Ā° and 360Ā°), and complete (360Ā°).