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The question provided is about a cyclic quadrilateral \( ABCD \), where the sum of opposite angles in a cyclic quadrilateral is always \( 180^\circ \).

Given:
- \( \angle A = (2x - 1) \)
- \( \angle B = (y + 5) \)
- \( \angle C = (2y + 15) \)
- \( \angle D = (4x - 7) \)

We know the following relationships:
1. \( \angle A + \angle C = 180^\circ \)
2. \( \angle B + \angle D = 180^\circ \)

Let's solve the system of equations step-by-step.

### 1. Equation for \( \angle A \) and \( \angle C \):
From \( \angle A + \angle C = 180^\circ \):
\[
(2x - 1) + (2y + 15) = 180
\]
Simplifying:
\[
2x + 2y + 14 = 180
\]
\[
2x + 2y = 166
\]
\[
x + y = 83 \quad \text{(Equation 1)}
\]

### 2. Equation for \( \angle B \) and \( \angle D \):
From \( \angle B + \angle D = 180^\circ \):
\[
(y + 5) + (4x - 7) = 180
\]
Simplifying:
\[
y + 4x - 2 = 180
\]
\[
y + 4x = 182 \quad \text{(Equation 2)}
\]

### 3. Solving the system of equations:
Now solve the system of equations:
1. \( x + y = 83 \)
2. \( y + 4x = 182 \)

Subtract Equation 1 from Equation 2:
\[
(y + 4x) - (x + y) = 182 - 83
\]
\[
3x = 99
\]
\[
x = 33
\]

Substitute \( x = 33 \) into Equation 1:
\[
33 + y = 83
\]
\[
y = 50
\]

### 4. Calculating the angles:
- \( \angle A = 2x - 1 = 2(33) - 1 = 65^\circ \)
- \( \angle B = y + 5 = 50 + 5 = 55^\circ \)
- \( \angle C = 2y + 15 = 2(50) + 15 = 115^\circ \)
- \( \angle D = 4x - 7 = 4(33) - 7 = 125^\circ \)

### Final Answer:
The angles are:
- \( \angle A = 65^\circ \)
- \( \angle B = 55^\circ \)
- \( \angle C = 115^\circ \)
- \( \angle D = 125^\circ \)

Thus, the correct option is **D**.

Would you like more details on how to handle such problems? Here are some related questions for practice:

1. What is the sum of opposite angles in any cyclic quadrilateral?
2. How can you verify if a given quadrilateral is cyclic based on angle properties?
3. How do you solve systems of linear equations with two variables?
4. What other geometric properties are unique to cyclic quadrilaterals?
5. How can angle bisectors be used in cyclic quadrilaterals?

**Tip:** Always remember that in a cyclic quadrilateral, the sum of the measures of opposite angles is \( 180^\circ \).

A cyclic quadrilateral ABCD has angles given as: ∠A = (2x - 1), ∠B = (y + 5), ∠C = (2y + 15), and ∠D = (4x - 7). Find the values of all four angles.

Learn how to solve a cyclic quadrilateral problem where angles are given as algebraic expressions. Understand the properties of cyclic quadrilaterals and how to apply the sum of opposite angles theorem to solve for unknown variables.

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