Adjacent Angles - Definition, Properties, Examples

Adjacent angles are an important part of geometry. These angles always lie side by side and share a common vertex. We can see many adjacent angles examples in real life, like the corners of a table or the hands of a clock.

So, what are adjacent angles? In simple words, we can define adjacent angles as "When two angles have a common arm and a common vertex, they are known as adjacent angles." These angles never overlap. Learning the definition of adjacent angles helps us understand shapes, lines, and different types of angles better.

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What are Adjacent Angles?—Overview

In geometry, adjacent angles are two angles that are placed side by side. These angles always have a common vertex and a common arm. The other arms of both angles lie on opposite sides of the common arm. This is why they are called adjacent, which means "next to each other" in mathematics.

When we study angles in geometry, we learn that adjacent angles can also be complementary angles if their sum is 90 degrees or supplementary angles if their sum is 180 degrees. Understanding what are adjacent angles is very important to solve problems related to shapes, lines, and measurements.

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Definition of Adjacent Angle

The definition of adjacent angle is: Adjacent angles are those angles that are always side by side and share the same corner point and one common side. These angles never overlap each other, as shown below in the diagram.

Adjacent Angles Examples in Real Life

We can see adjacent angles examples around us every day. These angles are very common in real life and help us understand the concept easily.

• Door Hinges: When we open a door, the door and the frame make two angles. These angles share the door’s edge and the hinge as their common side and vertex. These are adjacent angles.

• Open Book: When we open a book and keep it flat, the left and right pages form angles on both sides of the middle line (crease). These are also adjacent angles because they share the crease as the common side and corner point.

• Intersecting Roads: When two roads meet, they form angles at the crossing point. The angles that share the same road as one side are adjacent angles.

• Pizza Slices: When two pizza slices are kept side by side, they form adjacent angles at the center of the pizza.

• Clock Hands: When the hour hand, minute hand, and second hand point in different directions, they form adjacent angles at the center of the clock.

• Car Steering Wheel: The design of some steering wheels shows three sections that form adjacent angles meeting at the center point.

These simple adjacent angles examples help students see how this maths concept is used in everyday life.

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Property of Adjacent Angles

There are some important properties that help us understand and identify adjacent angles easily. These properties tell us how adjacent angles look and behave in geometry.

• Common Vertex: Adjacent angles always share the same corner point. This point is called the common vertex.

• Common Arm: Adjacent angles always have one common arm (side) between them.

• Do Not Overlap: Adjacent angles never cross or cover each other.

• Non-Common Arms: The other arms of both angles lie on opposite sides of the common arm.

• Can Be Supplementary or Complementary: If the sum of adjacent angles is 90°, they are complementary angles. If the sum is 180°, they are supplementary angles.

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How to Identify Adjacent Angles?

Now that we have learned what are adjacent angles, it is also important to know how to identify adjacent angles accurately. By checking a few simple things, students can easily find adjacent angles in diagrams and real life.

• Check for Common Vertex: The angles must share the same corner point.

• Check for Common Arm: The angles must have one common arm (side).

• No Overlapping: The angles should not cross or cover each other.

• Side by Side: The angles must lie next to each other without any space or gap between them.

If two angles do not follow all these points, then they are not adjacent angles. Both the common vertex and the common arm must be present to call them adjacent angles.

Adjacent Angles in Parallelogram

In a parallelogram, there are four angles. Some of these angles are opposite, and some are adjacent. Adjacent angles in a parallelogram are the angles that are side by side and share a common vertex and a common side.

It is important to remember that adjacent angles in a parallelogram are always supplementary. This means that if we add their measures, the total will always be 180 degrees.

So, in a parallelogram:

• Opposite angles are equal.

• Adjacent angles always add up to 180°.

Understanding adjacent angles in a parallelogram helps students solve many geometry problems easily.

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How Adjacent Angles are Different from Vertical Angles

Many of us might think that adjacent angles and vertical angles are the same, but they are actually different. Let’s understand how.

• Adjacent angles always have a common vertex and a common arm. These angles lie side by side and touch each other. They do not have to be equal in size.

• On the other hand, vertical angles only have a common vertex but they do not share any arms. Vertical angles are always opposite to each other and they are always equal in size.

So, the main difference between the two is that adjacent angles are side by side and may not be equal, while vertical angles are opposite and are always equal.

What Are Supplementary Adjacent Angles?

Supplementary adjacent angles are two angles that lie side by side, share a common arm, and their total is always 180 degrees. When two adjacent angles are supplementary, they form a straight line.

We can find supplementary adjacent angles on a straight line or in shapes like parallelograms. For example, if two angles are next to each other on a straight line, their sum will always be 180°.

Are Adjacent Angles Similar to Complementary Angles?

Adjacent angles and complementary angles are different concepts of geometry and have unique properties, such as:

• Adjacent angles are two angles that are side by side. They share a common vertex and a common arm. However, adjacent angles can add up to any number, not just 90°.

• Whereas, complementary angles are two angles whose sum is always 90 degrees. These two angles do not need to be next to each other or share a side. They can be anywhere in a shape or even in different shapes.

So, adjacent angles and complementary angles are not always similar. Sometimes, adjacent angles can also be complementary if their sum is 90°, but this is not always true.

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What Are Non-Adjacent Angles?

Non-adjacent angles are angles that are not next to each other. They do not share a common vertex or a common arm. For example, in a parallelogram, the opposite angles are non-adjacent angles. These angles can also sometimes add up to 180°, but they are not touching or connected directly, as shown in the diagram below.

Solved Practice Questions on Adjacent Angles

Question 1: Two adjacent angles on a straight line are given. If one angle measures 55°, what is the measure of the other angle?

Solution: On a straight line, adjacent angles always add up to 180°. So, the other angle = 180° − 55° = 125°.

Question 2: In a parallelogram, one of the adjacent angles is 75°. What should be the measure of the next adjacent angle?

Solution: In a parallelogram, adjacent angles are supplementary. So, the other angle = 180° − 75° = 105°.

Question 3: Two adjacent angles are in the ratio 2:3. If the smaller angle is 40°, what is the size of the bigger angle?

Solution: The ratio is 2:3.

If the smaller angle is 40°, then: 40° ÷ 2 = 20 (This is the common part). So, the bigger angle = 20 × 3 = 60°.

Question 4: On the edge of a book, one angle is 90°. What is the measure of the angle right next to it if they are supplementary?

Solution: If the two adjacent angles are supplementary, they must add up to 180°. So, the other angle = 180° − 90° = 90°.

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