In cyclic quadrilateral $PQRS,$
\[\frac{\angle P}{2} = \frac{\angle Q}{5} = \frac{\angle R}{6}.\]Find the largest angle in quadrilateral $PQRS,$ in degrees.

We are given that quadrilateral $ PQRS $ is a cyclic quadrilateral, and the following relationships hold between its angles:

\[
\frac{\angle P}{2} = \frac{\angle Q}{5} = \frac{\angle R}{6}.
\]

Let us introduce a variable to express all these angles in terms of a common factor. Define this common factor as $ x $. Therefore, we can express the angles as:

\[
\angle P = 2x, \quad \angle Q = 5x, \quad \angle R = 6x.
\]

In a cyclic quadrilateral, the sum of the opposite angles is always $ 180^\circ $. Specifically, we have the following relations:

\[
\angle P + \angle R = 180^\circ \quad \text{and} \quad \angle Q + \angle S = 180^\circ.
\]

Substituting the expressions for $ \angle P $ and $ \angle R $ in terms of $ x $, we get:

\[
2x + 6x = 180^\circ,
\]
which simplifies to:
\[
8x = 180^\circ \quad \Rightarrow \quad x = \frac{180^\circ}{8} = 22.5^\circ.
\]

Now, we can substitute this value of $ x $ back into the expressions for $ \angle P $, $ \angle Q $, and $ \angle R $:

\[
\angle P = 2x = 2 \times 22.5^\circ = 45^\circ,
\]
\[
\angle Q = 5x = 5 \times 22.5^\circ = 112.5^\circ,
\]
\[
\angle R = 6x = 6 \times 22.5^\circ = 135^\circ.
\]

To find $ \angle S $, we use the fact that $ \angle Q + \angle S = 180^\circ $:

\[
112.5^\circ + \angle S = 180^\circ \quad \Rightarrow \quad \angle S = 180^\circ - 112.5^\circ = 67.5^\circ.
\]

Thus, the angles of the cyclic quadrilateral are:

\[
\angle P = 45^\circ, \quad \angle Q = 112.5^\circ, \quad \angle R = 135^\circ, \quad \angle S = 67.5^\circ.
\]

The largest angle in the quadrilateral is $ \boxed{135^\circ} $.

### Do you need more details, or have any questions? Here are 5 related questions to deepen your understanding:
1. What is the property of opposite angles in any cyclic quadrilateral?
2. Can we generalize this method to any quadrilateral with similar angle relations?
3. How does the fact that the quadrilateral is cyclic affect the solution?
4. How do the angle relationships in cyclic polygons compare to those in regular polygons?
5. What happens to the angle measures if we change one of the initial angle ratios?

**Tip**: In cyclic quadrilaterals, always remember that opposite angles are supplementary (add up to $180^\circ$).

In cyclic quadrilateral $PQRS,$ \[\frac{\angle P}{2} = \frac{\angle Q}{5} = \frac{\angle R}{6}.\] Find the largest angle in quadrilateral $PQRS,$ in degrees.

Solve the cyclic quadrilateral problem where the angle relationships $\frac{\angle P}{2} = \frac{\angle Q}{5} = \frac{\angle R}{6}$ hold. Learn how to find the largest angle using geometry and angle relations in cyclic quadrilaterals. Suitable for Grades 9-12.

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