What are Adjacent Angles and Vertical Angles?

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Learn about Adjacent Angles and Vertical Angles: Understand their definitions, properties, and how they relate to each other in geometry.

What are Adjacent Angles and Vertical Angles?

Adjacent angles are two angles that share a common vertex (endpoint) and a common side. The term “adjacent” means “next to” or “lying close to.” In other words, these angles are side by side and have a common side. The non-shared sides of adjacent angles point in opposite directions. Adjacent angles do not overlap; they are separate angles that are connected by a common point.

Vertical angles are a pair of opposite angles formed when two lines intersect each other. When two lines intersect, they create four angles, and vertical angles are the angles that are opposite each other. They have the same vertex (endpoint of the angle) and are formed by two pairs of opposite rays (sides of the angle) extending from the vertex in opposite directions.

The important property of vertical angles is that they are congruent, meaning they have the same measure. If one vertical angle measures, for example, 50 degrees, then the other vertical angle formed by the intersecting lines will also measure 50 degrees.

To summarise:

• Adjacent angles share a common vertex and side, and their non-shared sides point in opposite directions.
• Vertical angles are opposite angles formed by two intersecting lines, and they have the same measure (they are congruent)

Examples For Adjacent Angles

Adjacent angles are angles that share a common vertex and a common side but do not overlap. Here are some examples of adjacent angles:

Example 1:

Let’s consider two lines, line AB and line BC. If the angles formed at point B are ∠ABC and ∠CBD, then they are adjacent angles because they share the common vertex B and the common side line BC.

Example 2:

Consider a straight line segment. If you choose any two angles on the same side of the line, they will be adjacent angles. For instance, if you have a straight line segment XY and you pick two angles, ∠AXY and ∠BXY, they will be adjacent angles.

Example 3:

In a rectangle, the angles at each corner are adjacent. For example, if you consider a rectangle ABCD, then the angles ∠ABC, ∠BCD, ∠CDA, and ∠DAB are all adjacent angles.

Example 4:

Imagine two intersecting lines, line PQ and line RS, crossing at point O. The angles formed at the intersection, such as ∠POS and ∠ROQ, are adjacent angles because they share the common vertex O and the common side line OQ.

Example 5:

In a clock, the minute hand and the hour hand form adjacent angles at each hour mark. For instance, at 3 o’clock, the minute hand and the hour hand form adjacent angles.

Remember, adjacent angles are always non-overlapping and share a common vertex and a common side, but they may have different measures.

Examples of Vertical Angles

Vertical angles are formed by two intersecting lines. They are opposite to each other and have equal measures. Here are some examples of vertical angles without diagrams:

Example 1:

Consider two intersecting lines, Line AB and Line CD. The angles formed at the intersection are vertical angles. Let’s call them Angle 1 and Angle 2. Angle 1 is the angle between Line AB and Line CD on one side of the intersection, and Angle 2 is the angle between Line AB and Line CD on the other side of the intersection. Vertical angles are always congruent, which means Angle 1 and Angle 2 have the same measure.

Example 2:

Suppose you have two intersecting lines, Line PQ and Line RS. At the intersection, vertical angles are formed. We can name these angles as Angle A and Angle B. Angle A is on one side of the intersection between Line PQ and Line RS, and Angle B is on the other side. Since they are vertical angles, their measures are equal.

Example 3:

Let’s consider two intersecting lines, Line XY and Line ZW. At the point of intersection, four angles are formed. Two pairs of these angles are vertical angles. Let’s call them Angle X and Angle Z, which are opposite to each other, and Angle Y and Angle W, which are also opposite to each other. The vertical angles have equal measures: Angle X is congruent to Angle Z, and Angle Y is congruent to Angle W.

Example 4:

Imagine two intersecting lines, Line MN and Line OP. At the intersection, vertical angles are formed. We can denote them as Angle M and Angle O. Both Angle M and Angle O have the same measure because they are vertical angles.

Remember, the key property of vertical angles is that they are always equal in measure, irrespective of the specific angles formed or the orientation of the intersecting lines.

How to Find Adjacent and Vertical Angles?

Finding adjacent and vertical angles involves understanding the relationships between angles formed when two lines intersect. Let’s define adjacent and vertical angles:

Adjacent angles:

Adjacent angles are two angles that share a common vertex and a common side but do not overlap. They are side by side and together form a straight line (180 degrees) when added.

Vertical angles:

Vertical angles are a pair of non-adjacent angles formed by the intersection of two lines. They are opposite each other and share the same vertex. Vertical angles are always congruent, meaning they have the same measure.

Here’s how you can find adjacent and vertical angles:

Step 1: Identify the intersecting lines

Start by identifying the two intersecting lines or the lines where the angles are formed.

Step 2: Locate the common vertex

Find the point where the two lines intersect; this point is the common vertex for the angles you want to identify.

Step 3: Identify adjacent angles

Adjacent angles share a common vertex and a common side. Look for two angles that are next to each other and share the same vertex and side.

Step 4: Identify vertical angles

Vertical angles are opposite each other when two lines intersect. Look for two non-adjacent angles that share the same vertex but are not side by side.

Step 5: Measure or determine the relationship between angles

You can use a protractor to measure the angles’ size, or if you know one angle’s measure, you can deduce the measure of the other angle using the properties mentioned earlier. Remember, vertical angles are always congruent.

Step 6: Verify the relationship

Check if the sum of adjacent angles forms a straight line (180 degrees). Also, verify that vertical angles are indeed congruent (they have equal measures).

Difference Between Adjacent Angles and Vertical Angles

Here is the major difference between the Adjacent Angles and Vertical Angles that will gives you a clear understanding about the Adjacent Angles and Vertical Angles.

Solved Examples on Adjacent Angles and Vertical Angles

Here are some solved examples on adjacent angles and vertical angles.

Example 1:

Consider two intersecting lines, line AB and line CD. The angles formed are:

Angle 1 (a): Measures 40 degrees.

Angle 2 (b): Measures 120 degrees.

Are angles 1 and 2 adjacent angles or vertical angles?

Solution:

Adjacent angles are two angles that share a common vertex and a common side but do not overlap. Vertical angles are a pair of non-adjacent angles formed by two intersecting lines. Let’s check the properties:

Angle 1 and Angle 2 share a common vertex (the point of intersection).

They also share a common side (the line CD).

Since both angles share a common vertex and a common side, they are adjacent angles. They cannot be vertical angles because vertical angles must be non-adjacent, meaning they do not share a common side.

Example 2:

Consider two intersecting lines, line PQ and line RS. The angles formed are:

Angle PQR (x): Measures 75 degrees.

Angle RST (y): Measures 75 degrees.

Are angles PQR and RST adjacent angles or vertical angles?

Solution:

Let’s check the properties:

Angle PQR and Angle RST share a common vertex (the point R).

They also share a common side (the line RS).

Since both angles share a common vertex and a common side, they are adjacent angles. They cannot be vertical angles because vertical angles must be non-adjacent, meaning they do not share a common side.

Example 3:

Consider two intersecting lines, line LM and line NO. The angles formed are:

Angle LMO (a): Measures 50 degrees.

Angle MOP (b): Measures 130 degrees.

Are angles LMO and MOP adjacent angles or vertical angles?

Solution:

Let’s check the properties:

Angle LMO and Angle MOP share a common vertex (the point M).

They also share a common side (the line MO).

Since both angles share a common vertex and a common side, they are adjacent angles. They cannot be vertical angles because vertical angles must be non-adjacent, meaning they do not share a common side.

In summary, understanding the properties of adjacent angles and vertical angles helps us identify and distinguish them without the need for diagrams.

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