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To solve this problem, we need to use the properties of equilateral triangles. Let's analyze the given diagram:

1. **Triangle ABC** and **Triangle ADE** are equilateral triangles.
2. Each angle in an equilateral triangle is \(60^\circ\).

### Step-by-Step Solution:

#### Step 1: Understanding the Angles

- Since **ABC** is an equilateral triangle, \(\angle ABC = 60^\circ\) and \(\angle BAC = 60^\circ\).
- Similarly, since **ADE** is an equilateral triangle, \(\angle ADE = 60^\circ\) and \(\angle DAE = 60^\circ\).

#### Step 2: Analyzing Angle Relationships

- \(\angle ABD = \angle ABC = 60^\circ\).
- \(\angle BAE\) can be expressed as the difference between \(\angle BAD\) and \(\angle DAE\), both of which are \(60^\circ\). 
- The angle \(\angle BAD = \angle BAC = 60^\circ\) and \(\angle DAE = 60^\circ\). 
- Therefore, \(\angle BAE = 60^\circ - 60^\circ = 0^\circ\). 

#### Step 3: Triangle ADE and Interior Angles

- Since \(\angle ADE = 60^\circ\), this means the interior angles in triangle ADE add up to \(180^\circ\).
- In the smaller triangle ABE, since \(\angle EAB = 60^\circ\), \(\angle AEB = 60^\circ\), the remaining angle \(\angle ABE\) also must be \(60^\circ\).

#### Step 4: Use Interior Angles to Solve for \(x\)

- Looking at the quadrilateral ABDC, we know that sum of angles in any quadrilateral is \(360^\circ\).
- We know \(\angle ABC = 60^\circ\), \(\angle ABD = 60^\circ\), and \(\angle DAB = 60^\circ\).
- We need to calculate the remaining \(\angle ADC\), with segments given as \(7x\) and \(\angle x\) within it. 
- Using the angle addition and subtraction in conjunction with known angles, we can equate:
\[
60^\circ + 60^\circ + (x + 60^\circ) = 180^\circ.
\]

#### Step 5: Solve for \(x\)

- From the above equation:
\[
x + 180^\circ = 180^\circ.
\]
- Subtract \(180^\circ\) from both sides to find:
\[
x = 0.
\]

#### Conclusion

Thus, based on the analysis of angles using equilateral triangle properties, the value of \(x = 0^\circ\).

Would you like more details or have any questions?

### Further Exploration Questions

1. What is the total internal angle sum of a triangle?
2. How do we prove that triangles are equilateral?
3. What other types of triangles have internal angles that are all equal?
4. How do properties of angles relate to the sides of a triangle?
5. What other shapes can have all sides equal besides equilateral triangles?

### Tip

When working with geometric figures, always start by identifying known properties and relationships, such as angles in equilateral triangles or supplementary angles, which can help simplify the problem.

ABC and ADE are equilateral triangles. Find the value of x.

Learn how to find the value of x in a geometry problem involving equilateral triangles ABC and ADE. This problem covers key geometric concepts, including the properties of equilateral triangles and angle relationships in polygons, suitable for students in Grades 8-10.

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