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To solve this problem, we will use the given angle measures and relationships. Let's summarize the given information:

1. \( m\angle GKL = 135 + 2x \)
2. \( m\angle JKG = 2x + 38 \)
3. \( m\angle JKL = 173^\circ \)

We are asked to find \( m\angle GKL \).

**Step 1: Understanding the relationships between angles**

From the diagram, \( \angle GKL \), \( \angle JKG \), and \( \angle JKL \) are part of a line segment extending from \( K \), forming a straight line. Hence, we can set up the equation:

\[
m\angle GKL + m\angle JKG = m\angle JKL
\]

**Step 2: Plugging in the given values and expressions**

Using the values from the problem statement:

\[
(135 + 2x) + (2x + 38) = 173
\]

**Step 3: Simplifying the equation**

\[
135 + 2x + 2x + 38 = 173
\]

Combine like terms:

\[
173 + 4x = 173
\]

**Step 4: Solving for \( x \)**

\[
4x = 173 - 173
\]
\[
4x = 0
\]
\[
x = 0
\]

**Step 5: Finding \( m\angle GKL \)**

Now substitute \( x = 0 \) back into the expression for \( m\angle GKL \):

\[
m\angle GKL = 135 + 2x = 135 + 2(0) = 135^\circ
\]

So, the measure of \( \angle GKL \) is \( 135^\circ \).

### Do you have any questions or would you like more details?

### Related Questions:

1. What are complementary and supplementary angles, and how are they used in geometry?
2. How can we solve for unknowns in angles when given more complex diagrams?
3. What is the significance of straight angle properties in solving angle problems?
4. How can algebraic methods be applied to find unknown angle measures?
5. How can you verify your answers in geometry problems involving angles?

### Tip:

When dealing with angles in geometry, always look for relationships like supplementary angles or vertically opposite angles to set up equations. These relationships are key to solving for unknowns.

Learn how to find the angle measure indicated in a geometry problem step-by-step. This problem involves using angle relationships and linear equations to solve for unknown angles. Suitable for students in Grades 9-10.

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