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The image shows a geometric diagram with three points labeled \(M\), \(P\), \(K\), and \(N\), forming two angles: \(\angle MPK\) and \(\angle KPN\). The problem states:

\[
\angle MPK = 2.6 \times \angle KPN
\]

We are asked to find the measures of \(\angle MPK\) and \(\angle KPN\).

### Solution:
Since \(M, P, N\) form a straight line, we know that the sum of the two angles on the same side of the line is:

\[
\angle MPK + \angle KPN = 180^\circ
\]

From the given information, we also know:

\[
\angle MPK = 2.6 \times \angle KPN
\]

Let \(\angle KPN = x\). Then \(\angle MPK = 2.6x\). Substituting this into the equation for the sum of the angles:

\[
2.6x + x = 180^\circ
\]

This simplifies to:

\[
3.6x = 180^\circ
\]

Solving for \(x\):

\[
x = \frac{180^\circ}{3.6} = 50^\circ
\]

Thus, \(\angle KPN = 50^\circ\).

Now, substitute \(x = 50^\circ\) back into the equation for \(\angle MPK\):

\[
\angle MPK = 2.6 \times 50^\circ = 130^\circ
\]

### Final Answer:
- \(\angle MPK = 130^\circ\)
- \(\angle KPN = 50^\circ\)

Would you like more details on this solution, or do you have any questions?

Here are 5 related questions to explore:
1. How do you derive the relationship between angles on a straight line?
2. What happens if the ratio between two angles changes?
3. How do you calculate angles in more complex geometric shapes?
4. How does this problem relate to the concept of supplementary angles?
5. Can you apply similar principles to find unknown angles in triangles?

**Tip:** Always check if two angles form a linear pair (adding up to \(180^\circ\)) when solving geometric problems involving straight lines.

Given that \(\angle MPK = 2.6 \times \angle KPN\), find the measures of \(\angle MPK\) and \(\angle KPN\).

Learn how to solve a geometry problem where two angles, \(\angle MPK\) and \(\angle KPN\), form a linear pair and are in a ratio of 2.6. This step-by-step solution shows how to use the supplementary angle theorem and algebraic methods to find the measures of both angles. Suitable for students in Grades 7-9.

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