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To solve this problem, we need to understand that quadrilateral \(ABCD\) is inscribed in a circle. In such a quadrilateral, the opposite angles are supplementary, meaning that the sum of the measures of opposite angles is \(180^\circ\).

Given:
\[
\angle A = 3x + 10^\circ, \quad \angle B = 3x + 20^\circ, \quad \angle C = 2x + 5^\circ
\]
We need to find \(\angle D\).

Since \(\angle A\) and \(\angle C\) are opposite angles, we have:
\[
\angle A + \angle C = 180^\circ
\]
Substituting the given expressions:
\[
(3x + 10) + (2x + 5) = 180
\]
Simplifying this:
\[
5x + 15 = 180
\]
Subtract 15 from both sides:
\[
5x = 165
\]
Divide by 5:
\[
x = 33
\]

Now, substitute \(x = 33\) back into the expressions for \(\angle A\) and \(\angle C\):
\[
\angle A = 3(33) + 10 = 99 + 10 = 109^\circ
\]
\[
\angle C = 2(33) + 5 = 66 + 5 = 71^\circ
\]

For opposite angles, \(\angle B\) and \(\angle D\) must also be supplementary:
\[
\angle B + \angle D = 180^\circ
\]
Substitute \(\angle B = 3x + 20\) with \(x = 33\):
\[
\angle B = 3(33) + 20 = 99 + 20 = 119^\circ
\]

Finally, solve for \(\angle D\):
\[
\angle D = 180^\circ - \angle B = 180^\circ - 119^\circ = 61^\circ
\]

So, \(\angle D = 61^\circ\), which matches the given answer. 

Would you like more details on any specific part of this solution, or do you have any other questions?

Here are some related questions:
1. What properties of a cyclic quadrilateral are essential for solving problems like this?
2. How do you verify if a quadrilateral can be inscribed in a circle?
3. What are the conditions under which opposite angles in a cyclic quadrilateral add up to \(180^\circ\)?
4. How can you derive the formula for angles in a cyclic quadrilateral?
5. What if one of the angle expressions was quadratic instead of linear?
6. How would the problem change if the quadrilateral was not cyclic?
7. Can you derive the same angle using the arc measurements in the circle?
8. What are some real-life applications of cyclic quadrilaterals?

**Tip:** Always double-check that the quadrilateral is cyclic before applying properties of inscribed angles and opposite angle sums.

Explore how to solve problems involving cyclic quadrilaterals using a step-by-step guide. Understand the properties of inscribed angles and opposite angle sums in a quadrilateral inscribed in a circle. Suitable for high school students.

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