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In the given image, we are asked to find the measure of \( \angle KNH \), and we are given two expressions for angles that are part of intersecting lines:

1. \( \angle JMG = (25x)^\circ \)
2. \( \angle HKM = (30x - 15)^\circ \)

### Solution:

These two angles are vertical angles, meaning they are equal. Hence, we can set their expressions equal to each other:

\[
25x = 30x - 15
\]

Now, solve for \( x \):

\[
25x - 30x = -15
\]
\[
-5x = -15
\]
\[
x = 3
\]

Now that we have \( x = 3 \), we can substitute this value back into one of the expressions to find the angle:

\[
\angle JMG = 25x = 25(3) = 75^\circ
\]

Since \( \angle KNH \) is adjacent to \( \angle HKM \) and these two angles form a straight line (they are supplementary), the sum of their measures must be \( 180^\circ \). Therefore:

\[
m\angle KNH = 180^\circ - 75^\circ = 105^\circ
\]

Thus, the measure of \( \angle KNH \) is:

\[
m\angle KNH = 105^\circ
\]

Would you like further details or explanations?

Here are 5 related questions to deepen your understanding:
1. How do we know that the two angles are vertical angles?
2. Can supplementary angles always be assumed when lines are intersecting?
3. What is the definition of vertical angles, and why are they congruent?
4. Could we have used a different method to verify the angle relationships?
5. What are other types of angle relationships when lines intersect?

**Tip:** Always check for vertical and supplementary angles when dealing with intersecting lines!

Find the measure of \( \angle KNH \) in the given diagram.

Learn how to find the measure of \( \angle KNH \) using the properties of vertical and supplementary angles. This step-by-step guide explains how to solve the problem using algebra and angle relationships, suitable for students in Grades 7-10.

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