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The problem presents a quadrilateral with labeled angles. The goal is to find the value of \( x \). 

The angles given are:
- \( 2x^\circ \) (bottom left angle)
- \( 45^\circ \) (top left angle)
- \( 40^\circ \) (top angle)
- \( 77^\circ \) (bottom right angle)

### Solution Steps
1. **Sum of Interior Angles**: The sum of the interior angles of a quadrilateral is always \( 360^\circ \).
   
2. **Set Up the Equation**:
   The equation based on the angles given:
   \[
   2x + 45 + 40 + 77 = 360
   \]

3. **Simplify the Equation**:
   Combine like terms:
   \[
   2x + 162 = 360
   \]

4. **Solve for \( x \)**:
   Subtract 162 from both sides:
   \[
   2x = 198
   \]
   Now, divide by 2:
   \[
   x = 99
   \]

### Answer
The value of \( x \) is \( 99^\circ \).

Would you like further details on any of these steps?

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Here are five related questions that build on this problem:
1. What is the sum of interior angles in a pentagon?
2. How would the equation change if one angle was given as \(3x\) instead of \(2x\)?
3. How can we verify if a quadrilateral is a parallelogram given only the angles?
4. What are the properties of quadrilaterals with supplementary opposite angles?
5. How can we find missing angles if a quadrilateral is inscribed in a circle?

**Tip:** For any polygon, the sum of the interior angles can be calculated using the formula \( 180(n - 2) \), where \( n \) is the number of sides.

Find the value of x in a quadrilateral with angles 2x°, 45°, 40°, and 77°.

Solve for x in a quadrilateral with angles 2x°, 45°, 40°, and 77° using the angle sum property. This problem explores key geometry concepts, suitable for students in Grades 7-9.

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