Class 5 - Angles Explained: Your Complete Guide To Types And Examples

When two rays come together at a same point, a geometric figure known as an angle is created. The vertex of the angle is the name given to this shared point. The sides or arms of the angle are the two rays that make up the angle. Consider an angle as the amount of rotation required to bring two rays together. Angles are measured in degrees. 360 degrees is the whole rotation around a point. An angle is represented by a caret or angle sign, which is written as ∠.

In this diagram, you can see two rays starting from a common point called the vertex. The angle is the space between these two rays. We can measure this angle in degrees to understand how open or closed the angle.

Table of Contents

• Six Types of Angles
• Angle Types Based on Rotation
• Pair of Angles
• Solved Examples on Angles
• Practice Questions on Angles

Six Types of Angles

Angles are classified based on their measure in degrees. Understanding these types helps us identify and work with different angles in geometry problems. Here are the six main types of angles:

1. Acute Angle

An acute angle measures between 0 and 90 degrees. It is always smaller than a right angle. Think of the acute angle as a narrow or pointed angle. Examples in everyday life include the angle formed between the hands of a clock at 2 o'clock or the tip of a needle.

2. Right Angle

A right angle measures exactly 90 degrees. It is represented by a small square drawn at the vertex where the two rays meet. Right angles are very common in geometry and everyday life. The corners of a book, a window, and a picture frame all form right angles.

3. Obtuse Angle

An obtuse angle measures between 90 and 180 degrees. It is larger than a right angle but smaller than a straight angle. An obtuse angle looks wider or more open than a right angle. If you open a door halfway, the angle formed is usually an obtuse angle.

4. Straight Angle

A straight angle measures exactly 180 degrees. When you measure a straight angle, the two rays form a straight line. This angle looks like a straight line when drawn. You can see a straight angle when a door is completely open or when you look at a road that goes straight ahead.

5. Reflex Angle

A reflex angle measures between 180 and 360 degrees. It is larger than a straight angle. When you have a reflex angle, you are measuring the bigger rotation between two rays. If you measure a regular angle and find it is small, then the reflex angle is the larger angle formed on the opposite side.

6. Complete Angle (Full Angle)

A complete angle measures exactly 360 degrees. This is one full rotation around a point. When you start from one ray and rotate all the way around until you come back to the same ray, you have made a complete angle. This is the total angle around any point.

Angle Types Based on Rotation

Angles can also be classified based on the direction of rotation. When we measure angles, we can rotate in two different ways:

Positive Angle (Counterclockwise)

A positive angle is measured in a counterclockwise direction from the initial ray to the terminal ray. In mathematics, counterclockwise rotation is considered the positive direction. When you measure an angle in the counterclockwise direction, the angle value is positive.

Negative Angle (Clockwise)

A negative angle is measured in a clockwise direction from the initial ray to the terminal ray. Clockwise rotation is considered the negative direction in mathematics. When you measure an angle in the clockwise direction, the angle value is negative.

Pair of Angles

When two angles are considered together, they form what we call a pair of angles. Different pairs of angles have special relationships and properties. These relationships help us solve many geometry problems.

1. Complementary Angles

Two angles are complementary if their measures add up to 90 degrees. For example, a 30-degree angle and a 60-degree angle are complementary because 30 + 60 = 90. Complementary angles do not need to be next to each other. They just need to have measures that total 90 degrees.

2. Supplementary Angles

Two angles are supplementary if their measures add up to 180 degrees. For instance, a 70-degree angle and a 110-degree angle are supplementary because 70 + 110 = 180. Supplementary angles often appear on a straight line.

3. Adjacent Angles

Adjacent angles are two angles that share a common vertex and a common side. However, they do not overlap. The other sides of adjacent angles are on opposite sides of the common side. When you see two angles next to each other sharing a ray, they are adjacent.

4. Vertically Opposite Angles

When two straight lines intersect, they form two pairs of vertically opposite angles. Angles that are opposite each other at the intersection point are called vertically opposite angles. These angles are always equal in measure. This is an important property that helps in solving many geometry problems

Solved Examples on Angles

Example 1: Finding Complementary Angles

Problem: If one angle measures 38 degrees, what is the measure of its complementary angle?

Solution:

We know that complementary angles add up to 90 degrees.

Let the complementary angle be x degrees.

We can write: 38 + x = 90

Solving: x = 90 - 38 = 52 degrees

Answer: The complementary angle is 52 degrees.

Example 2: Finding Supplementary Angles

Problem: Two supplementary angles have measures in the ratio 2:3. Find both angles.

Solution:

Let the angles be 2x and 3x (based on the given ratio).

Supplementary angles add up to 180 degrees.

So we can write: 2x + 3x = 180

This gives us: 5x = 180

Solving: x = 36

First angle = 2x = 2(36) = 72 degrees

Second angle = 3x = 3(36) = 108 degrees

Answer: The two angles are 72 degrees and 108 degrees.

Example 3: Identifying Angle Types

Problem: Identify the type of each angle: 45°, 90°, 135°, 180°, and 270°

Solution:

45° is between 0° and 90°, so it is an Acute Angle

90° is exactly 90°, so it is a Right Angle

135° is between 90° and 180°, so it is an Obtuse Angle

180° is exactly 180°, so it is a Straight Angle

270° is between 180° and 360°, so it is a Reflex Angle

Practice Questions on Angles

1: If an angle measures 56 degrees, what is the measure of its complementary angle?

2: Two supplementary angles are in the ratio 1:4. Find the measure of each angle.

3: Name the type of angle that measures exactly 180 degrees.

4: Two angles are vertically opposite to each other. If one angle is 65 degrees, what is the measure of the other angle?

5: Identify the types of angles measuring 25°, 150°, and 340°.

6: Three angles are complementary to each other. If two of them measure 30° and 20°, what is the measure of the third angle?

7: Explain the difference between vertically opposite angles and adjacent angles.

8: If angle A and angle B are supplementary, and angle A measures 123 degrees, find the measure of angle B.

9: Is it possible for an obtuse angle to be supplementary to an acute angle? Explain your answer.

10: When two lines intersect, they form four angles. If one angle is 72 degrees, what are the measures of the other three angles?