**What is the value of x in the figure below?**

Angles and their properties form the basis for determining unknown angle measures. Supplementary angles sum to 180 degrees, forming a linear pair. Complementary angles total 90 degrees. The Angle Sum Property states that the sum of angles in a triangle is 180 degrees. In the given figure, the angles are labeled as (x + 20) degrees, 60 degrees, and (x – 10) degrees. Applying the Angle Sum Property, we get (x + 20) + 60 + (x – 10) = 180. Simplifying and solving for x, we arrive at x = 50. Therefore, the value of x in the figure is 50 degrees.

## Unlocking the Secrets of Angles: A Quest for the Unknown

In the realm of mathematics, angles play a pivotal role, shaping our understanding of geometric figures and their relationships. They are the cornerstone of architecture, engineering, and art, guiding us in designing and constructing the world around us. As we embark on this mathematical adventure, we’ll delve into the intriguing world of angles and uncover their properties, with a particular focus on solving for the elusive value of *x*.

Imagine yourself as a master detective tasked with solving a geometric mystery. You’re given a sketch of an unknown figure with missing angle measurements. Your mission: to determine the value of *x* that completes the angle configuration. Fear not, for you’re armed with the knowledge of angle relationshipsâ€”a potent tool that will lead you to the truth.

## Angle Relationships: Understanding Supplementary and Complementary Angles

In the realm of geometry, angles play a crucial role in defining shapes and understanding their properties. When angles form certain relationships, they reveal valuable information that can help us solve complex problems. **Supplementary angles** and **complementary angles** are two such relationships that we will explore in this section.

**Supplementary Angles: A Linear Connection**

Imagine a straight line. It naturally divides a space into two parts, creating two rays that extend infinitely. When two angles share a common side and their sum equals 180 degrees, they are called supplementary angles. These angles essentially “complete” the straight line, forming a **linear pair**.

The relationship between supplementary angles is a fundamental property that is often utilized in geometric proofs and problem-solving. Understanding this relationship allows us to deduce unknown angles when only partial information is given.

**Complementary Angles: A Balance of Parts**

In contrast to supplementary angles, complementary angles have a different but equally significant relationship. Two angles are complementary if their sum equals 90 degrees. Think of it as a partnership where each angle contributes to complete a **right angle**.

Complementary angles frequently appear in geometric shapes, such as rectangles and squares. In these shapes, the interior angles at each vertex add up to 360 degrees, and therefore, every pair of adjacent angles is complementary.

## Related Concepts

Understanding the **Angle Sum Property** is crucial in this problem. This property states that the sum of the interior angles of a triangle is always **180 degrees**. This means that if we know the values of two angles in a triangle, we can easily find the value of the third angle.

Another related concept is **linear pairs**. Linear pairs are pairs of angles that are adjacent and form a straight line. This means that their sum is always **180 degrees**. Linear pairs are important in this problem because they can help us identify relationships between angles that are not directly given.

## Determining the Value of x: Unlocking the Secrets of Angles

When faced with an unknown angle in a geometric figure, determining its value can seem like a daunting task. But fear not, for there’s a secret formula that will guide you through this mathematical maze: the Angle Sum Property.

**The Angle Sum Property**

Imagine a triangle, with three angles labeled as A, B, and C. The Angle Sum Property states that the sum of the interior angles of a triangle is always equal to **180 degrees**. This fundamental principle holds true for all triangles, regardless of their shape or size.

**Applying the Property**

Now, let’s say we’re tasked with finding the value of *x* in a figure where some angles are given. Using the Angle Sum Property, we can set up an equation. For example, if we know that angles A and B measure 60 degrees and 70 degrees, respectively, we can write:

```
A + B + x = 180 degrees
```

**Solving for x**

To isolate *x*, we simply subtract the known angles from both sides of the equation:

```
x = 180 degrees - (A + B)
```

```
x = 180 degrees - (60 degrees + 70 degrees)
```

```
x = 180 degrees - 130 degrees
```

```
x = **50 degrees**
```

And there we have it! By harnessing the power of the Angle Sum Property, we’ve successfully unlocked the value of *x*. Now, you’re equipped with a secret weapon to conquer any angle mystery that comes your way.