In ΔJKL,  start overline, J, L, end overline*JL* is extended through point L to point M,  m, angle, J, K, L, equals, left parenthesis, x, plus, 9, right parenthesis, degreesm∠*JKL*=(*x*+9)∘,  m, angle, L, J, K, equals, left parenthesis, 2, x, plus, 9, right parenthesis, degreesm∠*LJK*=(2*x*+9)∘,   and m, angle, K, L, M, equals, left parenthesis, 5, x, plus, 6, right parenthesis, degreesm∠*KLM*=(5*x*+6)∘.   Find m, angle, J, K, L, .m∠*JKL*.

Given the triangle \( \Delta JKL \), where:

- \( \overline{JL} \) is extended through point L to point M,
- \( m\angle JKL = (x + 9)^\circ \),
- \( m\angle LJK = (2x + 9)^\circ \),
- \( m\angle KLM = (5x + 6)^\circ \).

We need to find \( m\angle JKL \).

### Step 1: Use the Exterior Angle Theorem
The exterior angle theorem states that the exterior angle of a triangle is equal to the sum of the two opposite interior angles. Here, \( m\angle KLM \) is an exterior angle, so we can write:

\[
m\angle KLM = m\angle JKL + m\angle LJK
\]

Substitute the given angle expressions:

\[
(5x + 6) = (x + 9) + (2x + 9)
\]

### Step 2: Solve for \( x \)
Simplify the equation:

\[
5x + 6 = 3x + 18
\]

Now subtract \( 3x \) from both sides:

\[
2x + 6 = 18
\]

Subtract 6 from both sides:

\[
2x = 12
\]

Now divide by 2:

\[
x = 6
\]

### Step 3: Find \( m\angle JKL \)
Now that we know \( x = 6 \), substitute it back into the expression for \( m\angle JKL \):

\[
m\angle JKL = (x + 9) = (6 + 9) = 15^\circ
\]

Thus, \( m\angle JKL = 15^\circ \).

### Final Answer:
The measure of \( m\angle JKL \) is \( 15^\circ \).

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Would you like more details or have any questions about this solution?

Here are five related questions to explore:

1. What is the value of \( m\angle LJK \) in the same triangle?
2. Can you explain the Exterior Angle Theorem in more detail?
3. How would the result change if the angles had different coefficients of \( x \)?
4. How can we check the consistency of the angle values in \( \Delta JKL \)?
5. How do supplementary angles play a role in this problem?

**Tip**: Always check whether you can apply the exterior angle theorem when dealing with extended lines from a triangle!

In ΔJKL, JL is extended through point L to point M, m∠JKL = (x + 9)°, m∠LJK = (2x + 9)°, and m∠KLM = (5x + 6)°. Find m∠JKL.

In this geometry problem, we use the Exterior Angle Theorem to solve for the angle m∠JKL in ΔJKL where the line JL is extended to point M. The angles m∠JKL = (x + 9)°, m∠LJK = (2x + 9)°, and m∠KLM = (5x + 6)° are given. Learn the step-by-step algebraic solution to find m∠JKL.

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