Adjacent Angles are the angles that have a common vertex, a common arm, and the rest two arms lie on either side of the common arm. Angles are particularly important in ge­ometry as they help de­fine and understand different geometric figures and their characteristics. Geome­try is a branch of mathematics that focuses on shapes, size­s, and angles. A key concept within ge­ometry is adjacent angles. Understanding adjacent angles helps us grasp the geometry of lines, polygons, and other shapes.

In this particular article, we are going to learn about adjacent angles, their definitions, their properties, and some examples and we will also practice some questions on it.

What is an Adjacent angle?

Adjacent angles are a pair of angles that share a common vertex and a common side but do not overlap. In simpler terms, they are angles that are side by side, touching at a single point, and not overlapping or intersecting.

The shared side is typically a ray, and the shared vertex is the endpoint of that ray. When two angles are adjacent and they form a straight line, also known as a straight angle.

Adjacent Angle Definition

The angles having a common vertex and a common side are known as adjacent angles. They are directly next to each other and do not overlap.

Examples of Adjacent Angles

Some of the Examples of Adjacent angles are mentioned below:

• Let’s conside­r a door with hinges placed at its edge­. The angles formed by the­ door and the frame at each hinge­ are called adjacent angle­s. These adjacent angle­s have a shared verte­x (the hinge) and a shared side­ (the edge of the­ door).
• When you ope­n a book and lay it flat, the pages form a straight line. The­ angles formed by the page­s on the left and right sides of the­ crease are calle­d adjacent angles. These­ angles share the cre­ase as a common side and verte­x.
• When two roads inte­rsect, the angles forme­d where they me­et are called adjace­nt angles. Each road serves as a share­d side for two adjacent angles.

Properties of Adjacent Angle

The properties of Adjacent Angle in ge­ometry are mentioned below:

• Common Vertex – The most critical property of adjacent angles is that they share a common vertex. The common endpoint is the point of intersection where the two rays meet.
• Common Side – Adjacent angles also share a common side. This means that one side of each angle is the same, and they form a straight line together.
• No Overlapping – Adjacent angles do not overlap; they are always next to each other. They have no interior points in common.

How to Identify Adjacent Angles?

We can identify adjacent angles, by using following steps [in no specific order]:

• Look for angles that share a common vertex and a common side.
• Check if the angles are formed by intersecting lines or rays.
• Ensure that the angles do not overlap.
• Confirm that the angles are next to each other, without any gaps between them.

Adjacent Angles in Parallelogram

In a parallelogram, two pairs of opposite angles are formed. Adjacent angles are those angles that share a common vertex and a common side, but they are not opposite angles.

Specifically, in a parallelogram:

• Opposite angles are congruent (they have the same measure).
• Adjacent angles are supplementary (the sum of their measures is 180°).

For example, in the diagram below, angles A and B are adjacent angles, as are angles C and D. Angle A and angle C are opposite angles, as are angle B and angle D.

So, if you know the measure of one angle in a parallelogram, you can find the measure of its adjacent angle by subtracting it from 180°.

Supplementary Adjacent Angles

Supplementary adjacent angles are pairs of adjacent angles whose measures add up to 180°. In other words, when two angles are supplementary, they form a straight line. However, in case of quadrilaterals such as parallelogramThis property is commonly observed in various geometric figures, including parallelograms, straight lines, and sometimes in triangles.

If you have a straight line, the angles on either side of the line are supplementary adjacent angles. For example, if angle A and angle B are adjacent angles on a straight line, then A + B = 180°.

In a parallelogram, opposite angles are congruent (have the same measure), and adjacent angles are supplementary. If angle A and angle B are adjacent angles in a parallelogram, then A + B = 180°.

Linear Pair Adjacent Angles

When adjacent angles form a straight line together, they are known as a linear pair. A linear pair of angles always adds up to 180°.

Complementary and Adjacent Angles

Complementary angles and adjacent angles are two different concepts in geometry.

Complementary angles are two angles that add up to 90°. In other words, if you have two angles, angle A and angle B, and they are complementary, then A + B = 90°. Complementary angles can be part of the same or different geometric figures; they don’t have to share a common side or vertex.

Adjacent angles are two angles that share a common vertex and a common side. They are next to each other and have a common side but don’t necessarily have to add up to any specific angle measure like complementary angles. Adjacent angles can be part of various geometric shapes, such as lines, polygons, or other figures.

However two angles can be complementary and adjacent as well. For Example if two adjacent angles sum up to 90° then we call that adjacent angles to be complementary as well.

Non-Adjacent Angles

Non-adjacent angles are angles that are not next to each other and do not share a common side or vertex. They are also referred to as “non-adjacent supplementary angles” when their measures add up to 180°. Non-adjacent angles can be found in various geometric shapes and configurations.

For Example, in a parallelogram, the angles that are opposite to each other are called Non-Adjacent Angles. Here, in the given parallelogram, A and C are Non-Adjacent Angles. Similarly, B and D are non-adjacent angles. In the figure attached below, angle I and angle F are also Non Adjacent Angles.

Conclusion

In conclusion, understanding adjacent angles is key in geometry. Recognizing angles that share a corner and a side helps solve geometric puzzles. Mastering this idea helps in both math and real life. So, keep spotting adjacent angles for better geometry skills and problem-solving!

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