Triangle L N M is cut by a bisector that forms a right angle with the base and splits the base into 2 congruent lengths. The lengths of sides L N and N M are congruent. Angle L N M is (6 x + 1) degrees. Angle N M L is (4 x minus 11) degrees.
What is the measure of ∠NLM?

m∠NLM =

We are given that triangle \( \triangle LNM \) has sides \( LN \) and \( NM \) that are congruent, which makes \( \triangle LNM \) an isosceles triangle. The angles at the base of an isosceles triangle are equal.

The angles are defined as follows:
- \( \angle LNM = 6x + 1 \)
- \( \angle NML = 4x - 11 \)
- \( \angle NLM \) is the unknown we need to find.

### Step 1: Apply the triangle angle sum theorem
The sum of the interior angles of any triangle is 180 degrees:
\[
\angle LNM + \angle NML + \angle NLM = 180^\circ
\]
Substitute the expressions for the angles:
\[
(6x + 1) + (4x - 11) + \angle NLM = 180
\]
Simplify the left-hand side:
\[
10x - 10 + \angle NLM = 180
\]
\[
\angle NLM = 180 - 10x + 10
\]
\[
\angle NLM = 190 - 10x
\]

### Step 2: Use the property of an isosceles triangle
Since \( LN = NM \), the base angles are congruent, meaning:
\[
\angle NLM = \angle NML
\]
This gives us:
\[
190 - 10x = 4x - 11
\]

### Step 3: Solve for \( x \)
Solve the equation for \( x \):
\[
190 + 11 = 4x + 10x
\]
\[
201 = 14x
\]
\[
x = \frac{201}{14} \approx 14.36
\]

### Step 4: Find \( \angle NLM \)
Now substitute \( x = 14.36 \) into the expression for \( \angle NLM \):
\[
\angle NLM = 190 - 10(14.36) = 190 - 143.6 = 46.4^\circ
\]

Thus, the measure of \( \angle NLM \) is approximately \( 46.4^\circ \).

Would you like a more detailed breakdown of any step? Here are five related questions:

1. How do you verify the properties of isosceles triangles in other contexts?
2. What happens if the triangle is not isosceles? How does it affect the solution?
3. Can you solve for \( x \) using a different approach in this problem?
4. How does the angle sum property apply to non-Euclidean triangles?
5. Why are the base angles of an isosceles triangle always equal?

**Tip:** When working with isosceles triangles, always remember that the two base angles are equal, which can simplify solving for unknown angles.

Triangle L N M is cut by a bisector that forms a right angle with the base and splits the base into 2 congruent lengths. The lengths of sides L N and N M are congruent. Angle L N M is (6x + 1) degrees. Angle N M L is (4x - 11) degrees. What is the measure of ∠NLM?

This problem involves solving for the unknown angle in an isosceles triangle where two angles are given as algebraic expressions. Learn how to apply the Triangle Angle Sum Theorem and Isosceles Triangle Theorem to find the measure of ∠NLM. Suitable for students in Grades 8-10.

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