What is Adjacent Angle in Geometry?

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 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.

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.

Solved Examples on Adjacent Angle

Example 1: Find the measures of two complementary adjacent angles if one angle measures 45°.

Solution:

If two angles are complementary, their measures add up to 90°. In this case, we know one angle measures 45°. To find the other angle, we subtract 45 from 90:

90° – 45° = 45°

So, the other angle also measures 45°. Therefore, the two complementary adjacent angles are both 45°.

Example 2: In a right triangle, one of the acute angles measures 30°. Find the measure of the other acute angle.

Solution:

In a right triangle, one of the acute angles is always 90°. The other two angles are the acute angles. If one acute angle measures 30°, we can find the measure of the other acute angle by subtracting 30 from 90:

90° – 30° = 60°

The other acute angle measures 60°.

Example 3: In a right triangle, one of the acute angles measures 40°. Find the measure of the other acute angle.

Solution:

In a right triangle, one of the acute angles is always 90°, and the sum of the measures of the acute angles is 90°. So, to find the measure of the other acute angle when one angle is 40°, subtract 40° from 90°:

Other acute angle = 90° – 40° = 50°

The measure of the other acute angle is 50°.

Example 4: If two vertical angles are formed by intersecting lines, and one of them measures 75°, what is the measure of the other vertical angle?

Solution:

Vertical angles are always congruent, which implies they measure the same. Given that one vertical angle measures 75°, the other vertical angle will also measure 75°.

Example 5: Two adjacent angles are complementary. If one angle measures 35°, find the measure of the other angle.

Solution:

If two angles are complementary, their measures add up to 90°. Given that one angle measures 35°, we can find the other angle by subtracting 35° from 90°:

Other angle = 90° – 35° = 55°

So, the other angle measures 55°, and the two adjacent angles are 35° and 55°.

Practice Questions on Adjacent Angles

Q1. In a straight line, if one angle measures 40°, what is the measure of the adjacent angle?

Q2. In a parallelogram, if one of the adjacent angles measures 60°, what is the measure of the other adjacent angle?

Q3. In a triangle, if one of the adjacent angles measures 45°, what is the measure of the adjacent angle?

Q4. Two angles are adjacent, and their measures are in the ratio of 3:4. If the larger angle measures 120°, find the measure of the smaller angle.

Q5. In a quadrilateral, the measures of two adjacent angles are 65° and 115°. Find the measures of the other two adjacent angles.

Adjacent Angles – FAQs

1. What are Adjacent Angles?

Adjacent angles are the angles having common vertex or common side. They are directly next to each other and do not overlap.

2. How are Adjacent Angles different from Vertical Angles?

Adjacent angles are angles that share a common vertex and side and are positioned next to each other. Vertical angles, on the other hand, are a specific type of adjacent angles formed when two lines intersect. They are opposite to each other and share a common vertex.

3. Can Adjacent Angles be Complementary or Supplementary?

Yes, adjacent angles can be either complementary or supplementary, depending on the sum of their measures. If the sum of two adjacent angles is 90°, they are complementary. If the sum is 180°, they are supplementary.

4. How are Adjacent Angles used in Real-World Applications?

Adjacent angles find practical applications in various fields, including architecture, engineering, surveying, navigation, traffic engineering, and trigonometry. They are used to ensure accurate measurements, design structures, and solve problems related to angles and directions.

5. What is difference between Adjacent Angle and Vertical Angles?

Adjacent Angles are different from Vertical Angles in the manner that in Adjacent angle one side is common while in vertical angles there is no common arms.

6. Are Vertical Angles always Congruent?

Yes, vertical angles are always congruent, meaning they have the same measure. If two lines intersect to form vertical angles, those angles will have equal measures.

7. Are Adjacent Angles always Congruent?

No, the adjacent angles are not always congruent. They may or may not be congruent.